OPTIMAL TOPOLOGY SELECTION OF CONTINUUM STRUCTURES WITH STRESS AND DISPLACEMENT CONSTRAINTS

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1 Th Svnth East Asa-Pacfc Confrnc on Structural Engnrng & Constructon August 27-29, 1999, Koch, Japan OPTIMAL TOPOLOGY SELECTION OF CONTINUUM STRUCTURES WITH STRESS AND DISPLACEMENT CONSTRAINTS Qng Quan LIANG 1, Y Mn XIE 1 and Grant Prntc STEVEN 2 ABSTRACT: Ths papr prsnts thr prformanc ndcs dvlopd by usng th scalng dsgn approach for assstng th slcton of optmal topologs for th mnmum-wght dsgn of contnuum structurs subct to strss and dsplacmnt constrants. Ths prformanc ndcs ar ncorporatd n th Evolutonary Structural Optmzaton (ESO) mthod to montor th optmzaton procss from whch optmal topologs can b dntfd. Exampls provdd dmonstrat that th proposd prformanc ndcs ar ffctv ndcators of matral ffcncy and can b usd to compar th ffcncy of structural topologs gnratd by dffrnt optmzaton mthods. KEYWORDS: volutonary structural optmzaton, dsplacmnt constrant, fnt lmnt analyss, prformanc ndx, strss constrant, topology 1. INTRODUCTION Th topology optmzaton of contnuum structurs has attractd consdrabl attnton n rcnt yars snc t offrs sgnfcant matral savngs than tradtonal szng optmzaton. Th Homognzaton mthod proposd by Bndsø and Kkuch [1] can b usd to optmz contnuum structurs by tratng lmnt dnsty as dsgn varabl. Th Soft Kll Opton (SKO) mthod outlnd by Matthck [2] s basd on th natur of bologcal growth. Th Evolutonary Structural Optmzaton (ESO) mthod prsntd by X and Stvn [3], Chu t al. [4] and Lang t al. [5] has found ts full us n ngnrng practc. It has bn found that usng dffrnt optmzaton mthods normally rsults n dffrnt topologs for th sam problm consdrd. In addton, vn usng on mthod, t s dffcult to dntfy th optmum n th optmzaton procss. As a rsult of ths, prformanc ndcs hav bn attmptd by rsarchrs to assst th slcton of structural topologs. Howvr, th form factors drvd by Burgss [6] ar only vald for smpl trusss and frams and not applcabl to contnuum structurs. Th prformanc ndx gvn by Qurn [7] dos not consdr any typ of constrant so that ts applcaton s vry lmtd. Th ndcator of matral ffcncy prsntd by Zhao t al. [8] s only vald for plan strss structurs undr a sngl pont load f th dsplacmnt constrant s consdrd. Ths papr dals wth th optmal topology slcton of contnuum structurs wth strss and dsplacmnt constrants by usng prformanc ndcs, whch ar drvd usng th scalng dsgn approach. Th ESO mthod for structurs wth strss and dsplacmnt constrants s brfly outlnd. Exampls ar provdd to llustrat th capablty of th proposd prformanc ndcs. Optmal topologs obtand by th prsnt study ar compard wth thos gvn by othr rsarchrs. 1 School of th Bult Envronmnt, Vctora Unvrsty of Tchnology, PO Box 14428, Mlbourn Cty MC, VIC 81, Australa 2 Dpartmnt of Aronautcal Engnrng, Th Unvrsty of Sydny, NSW 26, Australa 56

2 2. FORMULATION OF PERFORMANCE INDICES Th topology optmzaton problm of a contnuum structur can b statd as follows: mnmz w ( t ) W = n = 1 subct to σ σ or u u =1, m (2) whr W s th total wght of th structur, w s th wght of th th lmnt, t s th thcknss of th th lmnt that s also tratd as dsgn varabl, σ s th mum von Mss strss of an lmnt n th structur, σ s th prscrbd strss lmt, u s th magntud of th th dsplacmnt, u s th prscrbd lmt of th th dsplacmnt. To obtan th bst fasbl dsgn, th lmnt thcknss can b scald at ach traton so that th most actv constrant always rachs ts lmt [9-12]. Th stffnss matrx of a plan strss structur s a lnar functon of th lmnt thcknss. For a plan strss structur subct to strss constrants, by scalng th ntal dsgn wth a factor of σ, /σ, th scald wght of th ntal dsgn can b xprssd by (1) W s σ, / σ ) = ( W (4) whr W s th actual wght of th ntal dsgn and σ, s th mum von Mss strss of an lmnt n th ntal dsgn. In a sam mannr, th scald wght of th currnt dsgn at th th traton s gvn as W = ( σ / ) W (5) s, σ whr W s th actual wght of th currnt dsgn at th th traton and σ, s th mum von Mss strss of an lmnt n th currnt dsgn at th th traton. Th prformanc ndx at th th traton s dfnd by PI s = W / σ W (6) s s W = σ,w / vm, For structurs wth unformly dstrbutd matral dnsty, th prformanc ndx can b wrttn as PI s = σ V / σ V (7),, whr V and V ar th volums of ntal dsgn and currnt dsgn at th th traton rspctvly. For plan strss structurs wth dsplacmnt constrants, th prformanc ndx [11] s PI = / u V (8) d u V whr u and u ar th most crtcal constrand dsplacmnt n th ntal dsgn and n th currnt dsgn rspctvly. For plats n bndng, th stffnss matrx s th cub root of th plat thcknss. Th prformanc ndx at th th traton as prsntd by Lang t al. [12] can b formulatd as PI = u V (9) b 1/ 3 1/ 3 V / u 561

3 It can b sn from Eqs. 7 to 9 that prformanc ndcs ar dmnsonlss numbrs whch masur th matral ffcncy n rsstng th strngth falur or dflcton of a structur. Thy ar valuatd by th most actv constrant and th volums at ach traton. Snc prformanc ndcs ar rvrsly proportonal to th volum of th currnt dsgn, mnmzng th wght of a structur can b achvd by mzng th prformanc ndx n an optmzaton procss. It s notd that scalng th lmnt thcknss has no ffct on th optmal topology or on th prformanc ndx, but has a sgnfcant ffct on th wght of th structur and th actv constrant. Hnc, th lmnt thcknss s not changd n th modl at ach traton, but t can b changd n szng th obtand optmal topology to satsfy th actual prscrbd lmt. 3. EVOLUTIONARY TOPOLOGY OPTIMIZATION 3.1 STRESS CONSTRAINTS Th ESO mthod proposd by X and Stvn [3] s basd on th fully strssd dsgn concpt, n whch lowly strssd lmnts ar systmatcally rmovd from th structur to obtan an ffcnt dsgn. Th mum von Mss strss s usd as th lmnt rmoval crtra, whch s xprssd by σ < σ (1) RRk, n whch σ s th von Mss strss of th th lmnt, σ, s th mum von Mss strss of an lmnt n th currnt dsgn at th th traton and RR k s th Rcton Rato at th kth stady stat. All lmnts that satsfy Eq. 1 ar dltd from th structur. Th cycl of lmnt rmoval and fnt lmnt analyss s rpatd by usng th sam RR k untl no mor lmnts can b dltd from th modl at th currnt stady stat. An Evoluton Rato ER s thn addd to th RR k, whch bcoms RRk +1 = RRk + ER (11) Snc thr s no obctv functon and constrants nvolvd n th abov tradtonal ESO procdur, th optmal topology for th mnmum-wght dsgn cannot b dntfd durng th optmzaton procss. Ths problm s solvd by ncorporatng th proposd prformanc ndx PI s wth strss constrants n th optmzaton procss. 3.2 DISPLACEMENT CONSTRAINTS In th ESO mthod for structurs wth dsplacmnt constrants [4], lmnts wth lttl contrbuton to th structur stffnss ar rmovd from th structur to achv th mnmum-wght dsgn. Inffcnt matrals ar dntfd by th snstvty numbrs, whch ar dfnd by T α = { u } [ k ]{ u }, for sngl dsplacmnt constrant (12) = m T α λ { u } [ k ]{ u }, for multpl dsplacmnt constrants (13) = 1 T whr { u } s th nodal dsplacmnt vctor of th th lmnt undr th unt load corrspondng to th th dsplacmnt componnt, [ k ] s th stffnss matrx of th th lmnt, { u } s th nodal dsplacmnt vctor of th th lmnt undr th appld loads and th wghtng paramtr λ s chosn as / u u and m s th total numbr of constrants. Snc no obctv functon s usd to control th optmzaton procss n th ESO procdur 562

4 prsntd by Chu t al. [4], th optmal topology s dffcult to b dtrmnd. Th prformanc ndcs PI d and PI b prsntd can b usd n th abov ESO procdur to montor th prformanc hstory, from whch th optmal topology s asly dntfd. In th optmzaton procss, only a small numbr of lmnts that hav th lowst snstvty numbrs ar lmnatd from th dsgn at ach traton. Th Elmnt Rmoval Rato (ERR) s dfnd as th rato of th numbr of lmnts to b rmovd to th total numbr of lmnts n th ntal dsgn doman. 3.3 OPTIMIZATION PROCEDURE Th volutonary optmzaton procdur s gvn as follows: Stp 1: Modl th structur wth a fn msh of fnt lmnts; Stp 2: Analyz th structur for th appld load and unt loads; Stp 3: Calculat th prformanc ndx; Stp 4: Calculat th von Mss strss of lmnts or snstvty numbr for ach lmnt; Stp 5: Rmov lmnts wth low strss lvl or wth th lowst snstvty numbrs; Stp 6: Rpat Stps 2 to 5 untl th prformanc ndx s lss than 1 or constant n latr tratons. 4. EXAMPLES 4.1 EXAMPLE 1 Th ESO mthod for structurs wth strss constrants s usd to optmz th bam wth fxd nds as shown n Fg. 1. A concntratd load of 1 kn s appld to th top of th md span of th bam. Th dsgn doman s dstrtzd nto 9x3 four-nod plan strss lmnts. Th Young s modulus of matral E=2 GPa, Posson s rato v=.3 and lmnt thcknss t=15 mm ar assumd. Th Rcton Rato RR = 1% and Evoluton Rato ER=1% ar adoptd n th optmzaton procss. Th prformanc ndx hstory of th bam s shown n Fg. 2. It can b sn that th mum prformanc ndx s 8.54, whch mans that th scald wght of th ntal dsgn s 8.54 tms that th optmal topology as llustratd n Fg. 3 whl th mum von Mss strss of lmnts n th dsgn rachs th lmt. Fg. 4 shows th rgnratd fnal dsgn proposal gvn by Matthck [2] usng th Soft Kll Opton (SKO) mthod. Th prformanc ndx of ths proposal by usng Eq. 7 s found to b 1.51, whch s much lss than that obtand by th ESO mthod. Fg. 1 Dsgn doman E E E E E E E E Itraton E+2 Fg. 2 Prformanc ndx hstory PI s Fg. 3 Optmal topology Fg. 4 Dsgn proposal by Matthck [2] 563

5 4.2 EXAMPLE 2 Th dsgn doman of a cantlvr bams shown n Fg. 5 s modld usng 32x2 four-nod plan strss lmnts. A concntratd load of 3 kn s placd at th cntr of th fr nd whr a dsplacmnt constrant s mposd. Th matral proprts E=2 GPa, v=.3 and t=1mm ar usd. Th lmnt rmoval rato ERR=2% s mployd n ths cas. Th prformanc ndx hstory s shown n Fg.6, from whch t s sn that th mum prformanc ndx s 1.2 for th optmal topology prsntd n Fg. 7. Th prformanc ndx of th topology obtand by Chu t al. [4] usng th ESO mthod as llustratd n Fg. 8 s 1.11, whch s calculatd by Eq. 8. Th topology shown n Fg. 9 s prsntd by Suzuk and Kkuch [13] usng th Homognzaton mthod and ts prformanc ndx s found to b 1.4. It s clar that th optmal topology gvn by th prsnt study has a hghr ffcncy than thos obtand by othr rsarchrs. Fg. 5 Dsgn doman E E E E E E E E E Itraton 3.61E-1 Fg. 6 Prformanc ndx hstory PI d Fg. 7 Topology by prsnt study Fg. 8 Topology by Chu t al. [4] Fg. 9 Topology by Suzuk and Kkuch [13] 4.3 EXAMPLE 3 A smply supportd squar plat (2x2) undr a pont load of.4 N at th cntr s optmzd usng th prformanc ndx PI b wth th ESO mthod. Th ntal plat s modld usng 64 thrnod plat lmnts. A dsplacmnt constrant s mposd at th cntr of th plat. E= GPa, v=.3 and t=.1 mm ar adoptd. ERR=1% s usd n th optmzaton procss. It s sn from Fg. 1 that th mum prformanc ndx obtand by th prsnt study s 1.64, whch corrsponds to th topology llustratd n Fg. 11. Th prformanc ndx of th topology gvn by Atrk [14] as shown n Fg. 12 s It can b concludd that th ffcncy of matral layout n th bndng plat can b compard va th proposd prformanc ndx PI b E E E E E E E E E+ Itraton E+ Fg. 1 Prformanc ndx hstory Fg. 11 Topology by prsnt study Fg. 12 Topology by Atrk [14] PI b 564

6 5. CONCLUSIONS Thr prformanc ndcs hav bn prsntd for th topology optmzaton of plan strss structurs wth strss and dsplacmnt constrants and of plats n bndng. It s shown that th proposd prformanc ndx can b ncorporatd n any structural optmzaton mthod such as th ESO approach to montor th optmzaton procss, from whch th optmal topology of th structur can b asly dntfd. In addton, th ffcncy of structural topologs producd by dffrnt optmzaton mthods can b obctvly valuatd by usng th prformanc ndcs. ACKNOWLEDGMENT Ths papr forms part of a program of rsarch nto optmal topology and shap slcton n structural dsgn bng undrtakng at Vctora Unvrsty of Tchnology and th Unvrsty of Sydny. Ths program s fundd by th Australan Rsarch Councl Larg Grants. Th frst author s supportd by an Australan Postgraduat Award and a Faculty of Engnrng and Scnc Scholarshp. REFERENCES [1] Bndsø, M.P. and Kkuch, N., Gnratng optmal topologs n structural dsgn usng a homognzaton mthod, Comp. Mth. Appl. Mch. Engrg., 71, 1988, pp [2] Matthck, C., Dsgn n natur: larnng from trs, Sprngr-Vrlag, Brln, [3] X, Y.M. and Stvn, G.P., A smpl volutonary procdur for structural optmzaton, Computrs & Structurs, 49(5), 1993, pp [4] Chu, D.N., X, Y.M., Hra, A. and Stvn, G.P., Evolutonary structural optmzaton for problms wth stffnss constrants, Fnt Elmnts n Analyss and Dsgn, 21, 1996, pp [5] Lang, Q.Q., X, Y.M. and Stvn, G.P., Topology optmzaton of strut-and-t modls n rnforcd concrt structurs usng an volutonary procdur, ACI Structural Journal, 1999 (submttd). [6] Burgss, S.C., Th rankng of ffcncy of structural layouts usng form factors. Part 1: dsgn for stffnss, Proc. Instn. Mch. Engrs., Part C, J. Mch. Engrg. Sc., 212(C2), 1998, pp [7] Qurn, O.M., Evolutonary structural optmzaton: strss basd formulaton and mplmntaton, PhD Thss, Th Unvrsty of Sydny, Australa, [8] Zhao, C.B., Hornby, P., Stvn, G.P. and X, Y.M., A gnralzd volutonary mthod for numrcal topology optmzaton of structurs undr statc loadng condtons, Structural Optmzaton, 15, 1998, pp [9] Krsch, U., Optmal dsgn basd on approxmat scalng, Journal of Structural Engnrng, 18(ST4), 1982, pp [1] Lang, Q.Q., X, Y.M. and Stvn, G.P., Optmal slcton of topologs for th mnmumwght dsgn of contnuum structurs wth strss constrants, Proc. Instn. Mch. Engrs., Part C, J. Mch. Engrg. Sc., 1998 (accptd). [11] Lang, Q.Q., X, Y.M. and Stvn, G.P., Optmal topology slcton of contnuum structurs wth dsplacmnt constrants, Computrs & Structurs, 1998 (submttd). [12] Lang, Q.Q., X, Y.M. and Stvn, G.P., A prformanc ndx for topology and shap optmzaton of plat bndng problms wth dsplacmnt constrants, Structural Optmzaton, 1998 (submttd). [13] Suzuk, K. and Kkuch, N., A homognzaton mthod for shap and topology optmzaton, Comp. Mth. Appl. Mch. Engrg., 93, 1991, pp [14] Atrk, E., Shap: a program for shap optmzaton of contnuum structurs, In: Brbba, C.A.; Hrnandz, S. (ds), Computr Add Optmzaton Dsgn of Structurs: Applcatons, Southampton: Computatonal Mchancs Publcatons, 1989, pp