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Optimal design of wing structures with substructuring
compwrs & .5ham Fwgmm
Vol. IO.pp.m-910 Press W.. 1919. Prinled inGrealBrihin
OPTIMAL DESIGN OF WING STRUCTURES WITH SUBSTRUCTURINGt A. K. GOVIL,$ J. S. ARORAO and E. J. HAUL IbtcridsDivision, College of Et&wing The Universityof Iowa, Iowa City, IA 52242,U.S.A. (Receioe428 Auge~t 1978;rcceiwxf for publicutionI I January 1979)
into a stnxtd
opbibon
dgofithm.
1. -ON
recent paper[II, a generalmethodfor cplcdatins designderivatives of a.mcmaI mponsc quantities,such asstressesanddisp&ments,using&cconceptofsub sbuctWeswasdeveloped.Themethodutilizcsstate space of de* sen&ivity analysis in structural . . ideas . ptun~mn[2,3].Anewmethodofcakukt&design &i~ath with sub~bucturing wan inbgded into the gradient projection method for optimal structural design[4].Theln&hodwasapplkdto#eneralhussesand wasshowntpbemoreeBickntthanasimilarmethodU1 thatdoesnotuses&aWMn# Thepwpoacoft&ispaperis;Oextendapplkationsof theoptimalde&nmetbodwithsub&WW@tocompkxw-typestru&res.‘l%eopWaldes$nprobkmis formul&dinSectionxwheretb:colrtfunctionand f&re constraints for the wing des@ probkm an fk&Rd.Themainfeaturesoftheoj&naldesignaL In a
gkthlllwith~~Surmmaized.Detai)ed
derivationofthea&ithmisomittufintltkpaper,since it has been presented in a previous paper[4]. Extensive lrpplicrtioDsoftbeuWodanpresuWinSection3. Sevcaalnurn&alexa8r&sarecons&edandtheir optilllumsoWonsalecWsp8r&witlkotherresultsthat have been given in Refs.ET_ls].In order to have valid comp&ons,thcuniteelementnKMklsusedinI2efs. P-151 are also used in the present formulation to model the stnXtu@s. IWHMALaglcRaFwtNG~ TheoptiiDaldesignprobkmforidealiMwin$structures is de6ned as follows: IInd the design variabk assocktedwitheachekmentofawingstructure,such thatthetotalweightisatini&&,theequationsof e@&rium hold, and cons&a& on stress, dispkcement. andmembersixeansati&&Theclassoftwoand thret-dimensionalsWuctumlsystems considuzd herein is assumed to have a pteassioled geometric con&nation
and to be constructed of a known stn&tral material.
Furtheritisassumedthattbesestructurescanbeidealixcd using the following three types of finite ekments[lZ]: (i) truss ekmcnts of uniform cross-sectional arta (ii) constant strain Mngular (0 membrane elements of uniform thkkness, and (iii) symmetric shear panels (SW) or symmetric pure shear panels (SPSP). It is noted here that the SSP ckment diflers from the usual SPSP ekment in that it carries normal and shear loadsI6). Asanexampkofwingdesigqconsiderthestructure showninFii. l.Thestru&ueissynunetricwithrespect tothtx*-x~~one,soarlythcupperhplfofthestructme is modekd with CST, SSPISPSP, and ttuss ekments. Figure 2 shows the finite ekment i&&z&on of the stru&re. It consists of U nodes, 20 truss &men& (spar caps), 6ocsT ekments and 70 web eklmnts. The structurehasl2Odegreesoffreedornandl5fJliniteekments. whensparcapsarenotinchxkdinthefiniteekment ideal&ion, the structure has 130ekments. In analysis andoptimaldcsign,tkstructwemaybedividcdinto scveralsubstruc~s.sohltions forsevefalidealixations
ofthestructurearepresentedlaterintbepaper. Design variabks for wiqg structures are taken as cross-sec&al areas of the tn~s ekments and thicknesses of the CST and SSP/SPSP ekments. The design variabks for various types of ekments may also be combined into several groups to main&n symmetry in the structure and to reduce fabrication costs. The cost function, which is taken as the total weight of the structure, is an explicit function of the design variabks L
R(r)
Ilo(k)
NM(I)
(1)
where?‘P(r) = number of types of ekments in the rth subs&ucture;NGU) = number of groups in the &thtype of element; NM(i) = number of members in the ith group; b, = design variabk for ekmcnts of the ith group; &,= kngth of the jth truss ekment or surface area of meseerctl sqBportedby U.S. Army Anlmment Researcll and Devdapacat Commend, B&stics Research Labomtory, Aber- the jth CWSSPlSPSP ekment of the ith group; pc= weight density of ekments of the ith group; Z_. = total deen ProvingGrowk Wrylend. number of substructures. SW, Appiied l&clmks Depmtment,M.N.R. En* The CST, SSP and SPSP ekments are required to Cowe.fuwebad,lndie. 0Associate Frofessor. satisfy a stress criterion based on the Von Mises ~Professor. equivalent stress (also known as Maximum Energy of
A. K. GOVIL et al.
1
Fii. I. IS0 (130) Ekmcnt swept wing.
D&or&~ Theory(l61). According to this criterion, an eqtdvknt stns~ (u’) for CST w S!PWSP) &mm is givena6 ug=(o:,+u~-ollu~+3u,3
2
I/2 (2)
where 0; is the direct stress for truss elements or the
maximumvonMisesequ&kntstresscakula&dfrom eqn (2) for CST/!BP/W§P &ments and UP is an allowable stress. Displacolnent constlaints
at verious points of the
9lUCtUE~Wli~aS
where uij are the stress components at the point of in&test. The stress constraint for a typicai element of the wing structure is (3)
(4) where
zj is a calculated tipplacement and I,“ is, its allowablevalue. Furally, explicit bounds on design variabks are expressed as
(5) where bL and h” are lower and upper bounds on the ith design variable. The optimal desii formtdation for wing-type struc-
tures d&ed in eqns fl)-tS).Bts into the general formulation for strucarral opti&&n treated in Ref. [4]. Detailed derivation of the optimal design *aithm with substruchuingis presented there. Computptionolaspects of the algorithmare also discussed. Here, only the main features of the algorithmand pertinent expressions needs later are summarized: (1) The method is iterative and is based on the formula u=O,l,2..... b ("+1'=b'Y'+8b(Y';
(6)
Here Y is the iteration number, b”’ is a starting design and &“’ is a change in desii at the vth iteration. (2) The chan# in design 8b”’ is derived using KuhnTucker necessary conditions applied to a linearized prohlem and is given as a”‘= -*‘+a2
(7)
where T)is a step size and 66’ and sb* are given a~[41 (8) fig. 2. I50 (130) Ekmcnt swept wing element numbering.
6&Z= -W-‘/+2
(9)
Optimal designof
B = ATW-'A.
901
wiaa&uetnresWithsubstructuring
Optimaldesigns for four examples are obtained usitg the program of Ref. (17). TbeSe reSuIts are compamd withotherresultSavaikbkintheIikmtwe.Itshouldbe (11) noted that optimumd&M reporkd in Refs. (7-101ale
Here W is a positive &finite weighting matrix to be selected by the designer(6],A is a matrix whose columns are design derivatives of active constraints, and i is a vector containing values of active constraints at the current design point. The step size n is calculated based on a desired reduction r in the cost function at the start of the iterative process and is given as(2.4-61
(3) The de&n derivatives of active constraints are cakukted using the substructural data. These calculations proceed one substructure at a time. (4) Several members for diflerent substructures may beaSs@nedthesamedeGgnvariable. (5) All design variabk~ are allowed to vary in each deSigniteration. (6) The quantity
obtaincd~diirerentcomputersin~aentoperotioa
environments. Therefore comparisons of compu& times presented are only approximate.OptimumdeSigns withtbesubstNcturingmet&dureobtmneduSingan IBM 3tXMXH) compumr with doubk precision calculations. Optimaldesigns presented in Refs. (7,8] were obtai& using an IBM 36O-91compmer, which is roughly8timesfasterthanIBM3&6S@ivatecommunkation, IBM). Optimnl de+s preSented in Refs. (9,101were obtained using a CDC-7600compukr, which is approximately five times faster than the IBM MO 91(9, IO]. Exam& 1: 63 Member wing-carry-thmngh mnciun (Wcrs)
F~3&owsatrussidedix&nofthe63member wing-carry-tkoughstNcture(wCTs),nodalcoordirrptes (iinchesinparentheSeS),andthetwoloadingconditions.ThiSprobkmwrsorigi&yopti&edbyBerke and Khot(l21 and later by S&nit and MiuraI7,81and Dobbs and NeIson(ll]. The &ucture k modeled by 63 truss ekments and ham42 degas of freedom[l2]. . Intheoptimalde@nformuktionbys~ (J&‘l)2 = w6’=wtw) (13) Thestru&ueisdividedintn~~bypuris monitored. As the optimum is ppprosdled[Z], it is titk&gitatnodeS7-lO.The6int~cc&ktS of 31 member%:l-8,17-20,23-n, M-33,42&, 5& 59. required that pf?q2+0. Theremammg32mem&rSareintbeSu!oudSub&ucture. Tabk 1 gives member detMimm for the 1ArrucATKmANDcoy~~oFllDIIIuLf8 stnktuIe.subStNctm%ouehaSl8mtmiordqqcesof IntbisSectionthe~appli&ibtyandcomfreedomaudl2boundmyde#reeSoffreedmuSub put&nal etRckncy of the algo&hm for optimal Sulk- structum2haSl2inteSkrandl2boun&ydegnX!Sof tlaaldeSignwitllSttbS~iSdemonStrated,tk0Itph comparisonof the optimal des&ns for sevgil structures with known solutkns in the liiurep-15J. All Struck tures are divided into a number of subStructures for structural analysis and de&n SenSitivityanalysis. All calculations in the a@rithm proceed substntcture-wise. For detdled dkunuion of e@cientnumerkal impkmentation of the al&&m the reader is referred to Refs. 11,461. CONDITION 2 Acomputerprogmmbasedonthealgo&hmofRef.[4] and the formuktion preSented herein is developed in Ref. (171.The propom is automatic in the sense that deS@erneedStosupplydataaboutsubStructuresatKl starting values for the de&n variabks. The program continues execution until converpence is 0btaM or until a limit on the number of iterations is exceeded. The desipm has an option of performing a number of
“~tre~s-ratk’ design cycks at the start of the a4~o&hm[6].This is a poSitive feature of the program thatallowsthede&nertoobtainareasonabkdeGgnat the start of the a@ori&m[4].It must be noted however that even for the stress constrained design probkm the StreSS-ratiodeSi@cycles do not yield an optimumdesign for statica& i&terminate structures. When the structure is designed for multiple failure criteria, such as stress, displacement, bucllio& natural frequency, and OtherC0nStraintS, the stress-ratiodesigncycks produce a design that may be quite far from the optinuun. The authors have found that stress-ratio design cycles give a reasonabk initial design estimate calkd for in the Step 1 of optimal structural design algorithm with substructurinnl4.61. CAsVd. M?u.CE
FRONT l-30.0)
l-30.40)
VIEW
(-30.00)
(-30.120)
‘O& 12
TOPVIEW
+XI
IWE
I ) * COORDINATES IN INCHES
Fig. 3. 63 Member ~~&-carry-uuousb stmctuc.
A. K. GOVILerai. T&k 1.Memberlocationsfor63 membersWCTS
1
1.3
17
3.5
33
8.6
49
7.4
2
2.4
18
4.6
34
7.u
50
9.14
3
1.5
19
7.9
33
8.14
51
7.12
4
2.6
20
8.10
36
11.9
52
13.10
5
7.3
21
11.13
37
12.10
53
11.8
6
6.4
22
12.14
38
11.17
54
13.18
7
9.5
23
1.2
39
12.18
55
11.16
8
10.6
24
3.4
40
15.13
56
17.14
9
11.7
2S
5.6
41
16.14
57
15.12 5,4
10
12.8
26
7.8
42
1.6
58
11
13.9
27
9.10
43
1.4
59
3.6
12
14.10
28
11.12
44
5.2
60
9.8
3.2
61
7.10
13
lS,ll
29
13;14
45
14
16.12
30
3.9
46
5.10
62
13.12
15
17.13
31
4.10
47
3.6
63
11,14
16
18.14
32
7.S
48
9.6
Tabk 2.Results for 63 member WCTS I(rkr lhds 1
2 3 4 5 6 7 0 9 10 11 12 13 14 15 16 17 18 19 :: 23 24 25 26 27 28 29 30 31 32
37.4460 37.5610 52.7160 53.6230 23.8050 28.8300 17.1540 21.4020 26.2760 25.09so 8.8686 8.9586 23.4150 19.5970 5.1817 3.0207 37.0940 37.3540 0.0100 0.0100 0.0921
38.0780 36.1020 51.9230 54.2430 25.0690 28.2220 17.4610 20.7660 24.8100 26.9370 7.5854 9.0049 24.0470 20.4880 4.2966 2.9538 37.0600 37.1710 o.olOO 0.0100 0.0470 0.1048 1.1187 0.0312 2.9wO 1.3034 4.7730 0.8632 2.61109 2.7707 5.8034
0.0100 0.0100 0.0100 4.1985 0.9146 3.2270 0.0100 0.0127 8.0586 9.1997
Ud@t io lb NO.
I(u.
of
Activa
Colutrdnt
I
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 :: 54 55 56 57 58 59 60 61 62 63
optimum Deoipl in ln.L Robla
l(A)
5.6645 6.0951 5.6651 2.81S6 2.7665 2.6920 2.7716 5.7531 5.6651 16.0290 18.3970 11.4900 13.8640 11.73SO 6.1777 11.7930 13.9880 7.3SO8 7.7067 5.3699 0.021s 3.5992 9.7775 4.2407 0.1179 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 -4975.06
Constrdnt8
Violation
55 0.52 x lo-'
Problrl(B: 9.0427 9.6634 9.0430 8.4160 8.0605 7.9197 a.0632 9.264s 9.0417 24.3540 26.0850 19.3100 21.0440 17.0110 12.0530 18.4540 19.6470 6.5775 5.6535 0.4259 3.8917 6.1081 12.1060 5.7593 1.5657 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 -7 6117.38 23 0.34 x lo-=
3.68
0.91
21.64
39.b6
903
Optimaldesignof wingstructunswith&structuring Table3. Computationaltimes in seconds for 63 memberWCTS 1 Problm
Tote1 CPU
Tote1 CPU Time For Problem Setup
1
Streer-Ratio
1.987
l(B)
1.988
Design*
(2) + (3) + (4)
4
5
97.242
8.955 (lo)+ 11.641
Time
Opt-1
3
2
l(A)
Design
108.182
62.221
(13)+
75.850
*Inch&s CPU time requiredfor snidysis, sensitivity adysis and opfimhtion.
tNumbcrof stress-ratio designi&rationsperformed. freedom.Thus the entirestructwehas 12 boundary
degrees of freedom.
The structu~ is designed for two sets of constraints: (i) with stress constraints only (Problem l(A)) and (ii) with stress and torsional stillness constraints (FWbkm l(B)). Tbe torsional sti&ss constraint of Ref. [121is imposed by Emit& the relative displacementof nodes 1 and2intbex,dire&m.Nodesignvariablelinkingis used so there are 63 design variables for the structure. Design data for the WCT?Jare: material density, p f O.Mb/in’, modulus of elasticity (Titanium alloy) E= 16xlO‘psi,limitonreWivedispkcementofnodes1and 2 io the x~=direction=l.Oin., lower limit on crossse&mal areas - 0.01in’, upper limit on cross-sectional ar@a?i=noae, initial value of cro!Ss-sectionalaEas= 20.0ia’. Tabk 2 gives numerkal results for this exampk. For Probkm l(A), the set of active constmillts at the optimum in&&s stress in members I-15.17, 1%23-25,28, 5oaad5zuedWkadillgcoaditioo1,aadinmembers26, 30,32-3&3&49andu-56%underloadingcondition2. Minimmnsizeconatraintsonmembers19,2O,22and #)a3~eboective.ForproMem1(B),tbesttofadive . ums@wtsPttheoptimwnin&dcsstrcssinmcmhcrs 1-8,l&18,21,28,29,50 and 51, under loadingcondition
1,aadthetor&malstiffnessunderloadiqj0mdition2. Probkm l(A) rtquired 16 and P&km l(B) required 22 iteMionsoftheoptimaldesignaIgo&ntoconverge. The cost function historks are as follows: Robkm l(A): 4979.8, 4991.3, 4989.8, 4990.3, 4987.9, 4986.1, 4985.4, 4974.3, 4W.5, 4982.1, Ml.2 4980.6, 4979.2, 4977.9, 4976.7,4975.9,4975.1;and Problem l(B): 4977.5,5593.4,
6067.5, 7153.8, 6169.7, 6171.4, 6121.2, 6126.1, 6118.2. 6120.8, 611.7.9, 6139.4, 6147.1, 6144.7, 6144.2, 6138.9 6138.9,6134.2,6128.7,6124.0,6121.0.6118.5,6117.4. Wads of computa&maltimes with the substructurin8 algorithmare given in Table 3, sod Tabk 4 gives a comparison of results with other methods. The cost function is practicallythe same as repo&d in Ref. [7Jfor boththe probkms,whereasin compa&mwithRef.[12]it is about 1%lower.In compsr& withRef. [l 11,the tW weightachievedby the presentmethodis about1%lower for~l(A)and996~f~RoMcarltB).Finslly,it canbeseenfromTabk4thatthcreiscunderabk dll~iilcomputlaatiIllCUsiIlgtbC~dCsi$ll algwithmwithsUbs~
Example2: 18EIa~ntwingk~xbi~n
Fii4showsgeomebya1~ldime&msofw18 CkIWBtWiQ#bOXbC?ZMlWbdtWlOdii&coedi(ionr.TbG strwum issymmetricwidlItspect to fhe q-12 #me" Tbu&odytheapperlmlfoftbeboxbmmisitidw.
TWocasesfurthisexampkpfo&m8recxm&led In P&km 2(N), 5 busr, SC!JTand 8SSP dew&e wed.ForPr&km2@),SPSPelemm@musedinstd of SSP elements.In the optimaldesignformulntionwith subsbucturisgtheiddzedwingstnkctu~isdivided illtotwosub&uchWypprtitioaiapitrtnodes3and4. The firstsubstnsctrnecon&s of 2 trusselements(1.3). 2 CSTekments (1,2) and 3 SSPDPSPelements(1,3,6). the remainin ekments are in substructwe2. Both substructureshave6boundqdqrecsoffradom.Thefirst subsbucturchas no interiordegreesof freedom,whereas the second subsmlctufe has 9 inttxia degree!?of
Tabk4.Compn&onofrcsuksfor63membcrWCTS t
weight fn ibn
Probla l(A) Av. CPU Tke Per Cycle
Tote1 CPU T*
Prrrnt
4975.06 (la)*
6.08 (l.o)+
108.18 (1.0)'
Sckit-BKure [7] Jkrke-Khot [12]
4976.00 (14)* 5034.50 (50)*
6.43 (6.4)+ --
PO.47 (6.7)+ --_
Dobbe- Releon [ll]
5026.40 (ll)*
--
-_-
Problem l(B) Preeent
6117.4 (22)f
2.83 (l.O)+
75.85 (l.o)+
SctrPit-Miura [7] Berke-Kbot[12]
6120.9 (13)* 6159.3 (50)'
6.70 (113.9)~ -
87.51 (9.2)1 -
Dobbs-Nelson[111
6646.0 (20)*
ONamber of dssisncycks. tRelativecola& effort,
---
_-
I
A. K.
CST
c
SSP
et
al.
LOAD CONDITION 2 1DX103 Ibs
c> TRUSS 3
GOVIL
LOAD CONDITICN
1
Fig.4.I8Ekmcnt wingbox beam. freedom. Design data for this example are: p = 0.10lb/it?; E = 10X lb psi (Akminum alloy); Y= 0.30; 2,’ = 22.0 in. in hdhection; Op = 210 X lo’ psi; the bWerlimi~theupperhitandthestUtingareasfor ~NMekamnta are 0.02, I.0 and 0.98in’, respectively. For the remainingelements, tbe lower limit, the upper limit and tbe stating thicknessesare 0.02. 1.0 and 0.1%in., reSpcctivsly. Tabk 5 gives final results for both cases. The set of active con&mints at the optimum for Probkm 2(A) includee de&&on at node 5 in the x&ire&n under load&an&on I; Voa Miaesequivaknt stress for CST elansat 5 aad SSP ekments 3$ and 8 under loading coadith IandforSSPekments 1 and2underlonding co&ion 2; and mitt&m ekment size for truss ele-
ments 2-5 sod for SSP elements 6 and 7. For Probkm 2(B) the active comdrahts in&de ddktion at no& S in the x,-direction under load& an&ion 2; Van Mises equivakntstnssforCSTdement2iudSPSPd#nm:s 1 and 2 under loadingcondition 2; aod minimumelement size for truss ekments 2,3,5 and SPSP ekments 6 and 7. Problem 2(A) required 15and Probkm 2(B) requhd 9 opt&d design iterations to converge, The cost function histories are as follows: Probkm 2(A)do1.2, 424.4, 377.8.414.4,377.0,381.9,4fM.9,377.6,399.9,410.4,411).8, 406.4. 404.0, 403.0, 402.6, 402.7; and Probkm 2(B)355.9,359.7,354.1,360.8,360.7,360.2,3S8.9,358.8,357.!, 357.1. Details of computationaltimes with the sttbs&uc~
Tabk 5.Results for 18elementwingboxbeam
l-
WI 'robla 2(A)
3.o4.2
cst (k.1
3
s
6
1.2 3.4 5
0.004363 0.0s3070 0.037644
0.112730 0.053630 0.0364U
1 2 3 4
0.370soo 0.22#70 0.126540 0.120790 0.091634 0.02oaQO 0.02OWO 0.030954
0.086141 0.089aO 0.07462b 0.073004 0.073266 0.02OUM 0.02OOOO 0.024866
402.73
357.09
:
$sP For Prob. 2(A) kd
9 10 :;
6?sP lor Prob. 2(B)
13 14
:
tip.)
15 16
:
UeQbt ia lba. 8’ ii
lo.
t:iZ
O.low O.lom 0.1000 0.1000
of rtiva constraint
butraiat* vioktioll
0.10660 0. sbs22 0.1oooO
9 0.46xl63 44.16 l2os.00
Optimal design of wing structures with substructuring Table 6. Computationd times in seconds for 18element wing box beam TOM
cm
. Tiu
Totalcto
for
Dul#n*
w%+w
4
3
5
1
2
204)
0.847
l.SOS(4)+
9.800
12.152
20)
1.015
1.316(4)+
5.593
7.924 ,
*InChIdes CPU timereqnhdfor analysis, sensitivity anaiysis md optisniution. tNumber of stress-ratio design intcr&ions perform&
ofrcrubfor18ekmmtwingboxbcam T&k 'I.Coarporison Robla
’ uelgbt
Z(A)
Av.
Total Th
CPU The per cycle
ia lbr
CPU
Rewnt
402.7
(lS)*
0.65
(l.O$
12.15
(l.OF
Sdmit-Wure [71
403.0
(8)*
0.35
c4.3>+
3.37
(2.29
Itirsi [Q.lOI
402.7
(lS)*
0.06
(3.4)+
0.97
(3.2$
oa1lat1y
I141
_-
___
--
Cdlatly-Berke 1131
--
Z(B)
Roblar
Recent Scldt-Nlere171
357.1
(Q)*
0.62
(l.O)+
7.92
(1.09
357.8
(8)*
0.36
(4.6)+
3.43
(3.5)+
r9.101 cellecly-Berke (131
3a7.7
iurrl
w-e
___
(4)*
We_
-_
(193)*
___
-_
-em
389.8
cplletlY[141 *Number of de&a cycks.
tRektiWtzamphgeaort.
Tabk8.WLta
for 33&signvari&krectrnylprwi~
9brelldeerqndx2 llede
o=Penxt Iawl
L
of
mte
of bed ix
cod1thn
1
x3
lad
dirextioa
l
0.0
(kipd
bed bild:tial2
3 4 5 6 7 8 9
0.1656 0.4144 0.1426 0.3564 0.2311 0.3274 o.u77
0.2996 0.3745 0.2S76 0.322l 0.2367 0.2958 0.2127
10
0.2943
0.2659
11 a2 13 14
0.0978 0.2445 0.0381 0.0953
0.1768 0.2209 0.0689 0.0861
906
A. K. GOVIL eraL 0
TRUSS
14
Fig. 5.33&signvuiabk rtctmguhr wing.
Table 9.RCSU~~S for 33 design~a&&
E 21 22 23 24 BS? (in.)
:: 27 28 29 30 31 32 33
1 2 3 4 5 6 7 8 9 10 11 l2 13 14 1S
Ic g
W&&t lo. of hethe In lba.Comtrdnts Msx. Constraint Viohtion
:. -
11~~112 I16b1112
recw
0.08067S 0.059261 0.0414fP 0.064171 0.031335 0.047776 0.026171 0.03S431 0.0111733 0.028262 0.010 0.023174 0.010 0.010 0.010 100.344 10 o.S8xlo-7 102.70 17.46
wiry
0.0?))019 0.0727lb 0.041699 o.e6sm3 .0.029265 0.010s10 0.022464 0.04077s 0.014S46 0.031025 0.010 0.0131SS 0.010 0.010 0.010 100.212 15 o.'72atlo-7 102.70 1.94
901
Optimal designof wing structureswith substructuring
algorithmarc given in Table 6. Table 7 gives a summary cumpksun of results with uthcr methods. Compared to R&L [7,9], the cost function is practicaliy the same, whereas fur Prubkm 2(B) there is a reduction of abuut 9%, compared tu Rcfs. (13, IS]. The present method shows 0 computinptime advantage of u factor of two to fuur fur this relatively small scale problem. EuwJPfiw&; ~~~~~~
us
25x lo’psi. Fur truss elements biL= 0.010in2, hii0= 1.5in2.and the starting design is 2.0in2. Fur the rcmaining elements btL = 0.01in, br” was not spccihd and the starting d&n is OMin., except for CST ckunents 1-12 forwhichitisO.oBin.Thae~trwo~conditions forthestruc~whkharegiveninTabk8.Alsudesign variabk Iii is used for CST ekmcnts.
*
for a ~~~~.~t~&n~~T~9~~~u~s~,~hiss~~~~~~ xpf2 p&meand currcspunds to the wing middle surface. ~n~~~~l~~~~~~4~~n. Both designs arc usabk, since all cuj+nts are Tks,oniytheuppcrhaifoftherectmq&rwingis mud&d, usii@12 truss &ncuts to represent spar cap& satiskd. Also, the wci@ts at the twu rtembc~ arc 12 CST elements for the skin, and 15 SSP elements for practically the same. Huwzvcr, the final desipas arc somewhat d&rent and the vahms of m’b a~ quite the vceical webs. In m design formuMun with substructuriag,the different. The value d Ii%& 8t the 42nd ituatiun is muchsmauerthanattht15thiWatilm.Furtk.tbecost idc&ed*&ucturcisdividcdintothrccsubstructures by parti&& it at nodes 5-6 and 9-10. Truss func&misveryncaritsoptimumvalpeafWr15itm+ ekmcnts l-4, CST ckments Id and SSP ekncnts l-4 tions, but it takes 27 man iterations to signifk&y reduce the value of iWj2. This behaviur is oberved in aud13formslWtructurclandtnlsse~nts5-8,cST &mm&s 5-8 pad SSP elements 5-8 and 14 form sub ~ys~~~~~~~tf~~ &mar. S~Z~~~~~~~~S~~ ~~tof~~~at~~~~s 3,~~~~~~~~S~f~. ~~~~s~at~~4~~x~ ~~~~~rnf~~~~~~ under luad& cor&iun 2; the Van Mises eq&vaknt and third s&tr~ues are 6, 12 and 12, rcspetively. strcssinsSP&mcnts land3,undertomiil@cuxiMun Ikaign data for the probkm arc: p*O.lOlb/in’, E= 10.5X lobpsi (Ahtminumalloy), v T 0.30, zj’ = 211.0in. 2; and the minimumelement size fur truss elements 11 in the x3 dir&ion at nudes 13 and lrlf and 0: = and 12,CST ckmcnts 11and 12Mesignvariable 16).nnd
1.262
19
33
0.206
5%
z:
2.380 0.910
z 36
0.363 0.431
33 ii
2.013 2.393
z
0.302 0.144
1.764
26
:;
0.269 0.395
0.727 1.366
fz
1.297 1.906 0.570 1.196
z 3l 32
tlz 43 44
X:S 0.169 0.116
ii 3s
0.646 0.402 0.396 O.lS4 0.3ll 0.462
0.363
. 19
20 21
1.02s 0.3% 0.643
it
1.374 0.825
24 23
0.264 0.663
E s :
z
0.651 0.224 1.092 0.5l6
z
OAS
3:
O-SOB 0.175
": 44
i%i O.l33 O.pB6 0.231 0,566
A.
908
K. GOVIL et al.
SSP elements 11, 13. 14 and 15. The total computing time with the present algorithm, which includes initial set-up time (2.09sec). 4 iterations of stress-ratio design time (3.53 set), and 15 iterations of design time (17.69 set), is 23.31 sec. In Refs. [9, IO], the CPU times reported are 0.19 se&sign iteration and 3.14 for 15 design iterations. Comparing the CPU time per design iteration and total CPU time with Ref. [lo], it is observed that the substructuring formuiation yields a reduction of factors of 6.4 and 4.5, respectively. Lastly, the cost function history (in Ibs) for this problem is: 64.48, 8255, 99.79, 104.31, 104.61, 102.07. 101.10, 100.80, lOtl.M), 100.18, 100.12, 100.36, 100.36, 100.35, 100.36,100.34. The optimum cost function is pm&ally the same as in Refs. [9, lo]. It is noted that for this problem components of I%’ and 86’ associated with design variabks for ekments 11 and 12 (truss) and 31-33 (SSP) are equal to xero after the 4th iteration. Hence, in the subsequent iterathms these design variabks are kept fixed [6]. ErMlplr 4: 150 (130) dcrnat swept wing F-s 1 and 2 show geometry, dimensions and element numbering of the ideal&d swept wing structure. As noted ear&r, Only the upper half Of the swept wing is modekd,using2Otntssekmentstorqnesentthespar caps, 60 CST ekments to qresent the skin, and 70 SSP
elements to represent the vertical webs. The wing without spar caps is referred to as Probkm 4(A)and with spar caps (truss elements) as Probkm 4(B). in this example, extensive design variable linking is employed (see Table 11). The number of independent design variables describing the spar cap, skin and web are 14,ll and 7, respectively. For Probkm 4(A) there are eighteen designvariablesand for Probkm 4(B) there are 32. In the optimal design formuktion with substructuring, the idea&d swept wing structure is divided into two substnktures by partitioning at nodes 17-20. The first substMcwn consists of eight truss ekments (l-4,1 l-14), twenty four CST elements (l-24), and hvettty-flve SSP elements (l-4, 11-14, 21-24, 31-34, 41-49). The remaining ekments are in substructure two. Both substructures have 12 boundary degrees of freedom. The fkst has 36 interior degrees of freedom, whereas the second has 72. LIesii data for this example are: p = 0.0% lb/m’, E = 10.6x lo” psi (Ahuninium alloy); Y= 0.30, z,” = 60.0 in. in x&ire&on, and uia = 225 x ldpsi. For truss ekments 6,‘ = 0.01 in’, b,” = 1.50in2, and starting design is 0.02 in*, and for tbe remaining elements b,‘ = 0.02 in. and br” are not specifkd. For Probkm 4(a) the starting desii is 0.20 and 0.10 in. for CSI‘ ekments l-24 and 25-60 and 0.2Oin. for all SSP ekments, respectively. Thereare twoloedinpconditioasforthestructure,~given in Tabk 10.
T&k 1I. Results for 150 (130) element sweptwing
.1
1
3 4 ii :
1: ll
cm (to.)
12 l3 14 15 16 17 16 19 if 22 23 24 25 26 ii 29 ii! 32
: 4 5.6 7.8 9.10 ll 12 13 14 15.16 17.18 19.20 l-6 7-12 u-la 19-W 23-36 37-u m-60 l-4 3-10 11-14 13-20 21-24 r-z 3340 41-49 soda 59-70
0.203wo
0.177250 O.lwtso 0.1292% 0.11m10 0.094651 0.02oam 0.632631 0.02OOoO 0.033091 0.046713 0.222400 0.0979ao o.O9a411 0.010172 0.035310 0.061103 O.l6W 2462.34 13 o.*a~lo-~ 33.12 2647.00
O.OlOOO8 O.OlWW 0.01OoOo o.olOooo o.olmoo 0.01moo o.olO@6o 1.5OoOfM 0.637270 0.339730 0.716llO 1.0029Oo 0.369730 o.OlcmO O.lRZ(IO 0.171290 0.13174Q 0.123uo o.o99201 0.067910 0.02W 0.032648 O.o2oooo 0.03u125 0.017229 0.228710 0.11sa60 0.064393 o.os3397 0.033617 0.06a786 0.0993% 2442.% 22 0.w x104 35.11 3263.00
909
Table 11 gives numericsi results for both aiscs of this exampk.ThesetofactivecolIs~attBtoptimumfor Probkm YA) includes van Mises eqltivaknt stress for CsT&neats8,14and#)underl&ingconditkn1, s!@cIcments20,21,30,S8,and61u&rIoadilQumdition 1, and for SSP ekmellts 3, 11, and 42 under Ioadiq comMkn2,andlllhimumekment*forcsTekments 4%0 and SSP eIements5-10; Problem 4(B)was the same as for Frobksn 4(A) with addith of minimumarea for ~eiclneatsl-l0,19~2O,Pnd~~rveafor~ss eIemuIlt11. ~~~~~kf~~~~~~ in I&. f7, Ial, with the exception of the stress con-, saniatforSSPakmcdllamieri~conditioa2.Itis not#dthatnoncofthctipdelkcthconstraiatsart active. The d&but&~ of mat&l achiewd by the pMIltmCthodisappKhl@lythCMCaSiURCf.[71, cxccptforthcspnrcaps.TheoptimuJndesiimportcdin Tabk 11 is obtahl in 15 design iterations for both uurcs.ItisnotcdthatforRobkmya),a~desienis foundattbc1Othiterahwitbaweigbtof2469361b. wbicb is within 03% of the minimum achieved. For qtoMcm4tB)~ausabkdes@nisfoundatthe7thitera~~~~~~59.52lb, which is within 0.7% of
~_f~~~~~lS~~~~s~~~,~~~~~s~~f~: Probkm YA)-2340.1, 2432.7, 24715, 2460.1, 2459.8, 2457.3, 2452.7, 2467.4, 2463.2, 2469.4, 2467.0, 2465.4, 2444.2, 24fi3.2, 2462.7, 2462.3; and problem rySH440.8, U13.4, 2465.1, 2422.1, 2429.8, 2455.0, 2459.5, 24S7.0, 2454.1, 2448.7, 2443.9, 2443.4, 2442.7, w2.ig2442.5, 2442.1. DctaiIsof computationaitimes with the subst~cturing Table 13.Compa&onof
are given in T&k 12. Table 13 gives a compa&m of nsultswithotbcrmethods.Thccostflmctioa,ascompared to Rd. [9], is pm&ally the s8mc for botb cases. Itisthesrmefor~~YA),oscolnprradto~.Ttl, butabwt0.9%IessforPrdhmYB).Comqtingtimcby thepreseatmetbodisagainafactorofhvotoanordeiof m@tudelowertlumothermethods. Lastly, the swept wing is divhl into five sub&wtures by parthnii it at nodes 9-12.17~20.25-28 and 33-M. Nodes 41-44 are also umsidcred as boundwy ~S*W~~~,~~~~~~~~~~~~S*
~~nrn~~~~~~for~ ~~~A)~~).~~~PU~is~~~ same.TbisisdueotothcfactthattberotainPRIberof caIcuiptions with two ami five substructures is approximatelythe same.
sensithity analysis[2,3] and by incorporrdrm of subS~~~~~~.~U~cal-s~~htbcpnperarc~t inthcdesignoflwgcscakS-.Basedoat&
nsultsofthisnsearcb,itisconclu&dthatagencral purpoaopthUddC&llcomputerCodeWitbslIbstructlEin8 may be developed for a large class of strwtmkl design probkms,
fesldts for 150 (130) &men~ Swept wise
f
Problr
I -ight
in
Pr-t Schrit-Miura atrzi
[ 7]
I9.101
lbe
2462.3
(lS)*
2462.8 2461.8
Sclmit-Hiura Rlzri
r9.101
[71
I Tot&l CPU Time
4.61
(l.O)+
82.12
(LO)+
(8)*
2.53
(4.4)+
21.50
(2.1))
(16)*
0.75
(6.5)+
13.19
(6.4)+
ProblPresent
4(A)
CPU Nu per cycle
AV.
4(a)
2442.1
(U)*
4.94
(Lo)+
88.37
(l.o)+
2460.8
(9)*
3.69
(6.0)+
34.65
(3.1)+
2445.0
(16)*
1.16
(9.4)+
20.21
(9.2)+
‘Number of desii cycks. tR&tive computing &Tort.
910
A. K.
GOVIL et al.
1. I. S. Arora and A. K. God. Design sensitivityanalysiswith substructure. 1. &ng Mech. JXu. Proc. of ASCE. 103 (EM4), 537-548 (Aug. 1977). 2. E. J. Haug. Jr.. K. C. Paa and T. D. Streeta, A computathlalmatlwJdforoptimalrtrucanldasign:!piecewise uniform structures. ht. I. Namer. M&h. Engng 5, 171-184 (1972). 3. E. J. u and J. S. Arora, De+ sensitivity analysis of elastic iwcbicd systems. Coarpmter Methods in Applied Mechanics and EN&W& Vol. IS, pp. 35-82 (19%). 4. J. S. Arora and A; K. God. an etlkkt method fur optimal structural design by substruenrrinl. comput. StIwcttf#s 7(4). 507-515 (1977). 5. J. S. Arota and E. J. Haug, Jr., EtIkicnt optimal design of structuresby ganaahd swpcst &swat pmlramming. ht. /. Nunw Met. .Qtgng lU4), 747-766 (1976); and IU6), 14XL1426 (1916). 6. A. K. God, J. S. Arora aml E. 1. Haug, Substrwturintl
meulodsfnrdesign~aadysisandsmictmalopti&don. T~~.No.34(&oPh.D.Diswwbnofthe firsta&or)* kbt&ab Dildoas, cone@of E?&eIriqt, TbaulBksityofIowsIowaCity,lewr(Aug.1ppT). 7. L. A. S&nit, Jr. and H. t&n, &edm&~ Cvts for E.#ciant Skctuui Syntksk NASA CR-2552, University of
Califarnia, Los A&es, CA %I024(Mar. 1976). 8. LA.SabmitaodH.ffiAnew~ad~sisbn.
on optimatity criteria. Pm. of AIAA/ASME/SAE 17th s)NCtures, Stmctund @namics and Material Conf., pp.
448-462. King of Prussia, Pennsylvania (5-7 May 1976).
10.P. Rid, Optinizcltionof Structures with Complu Constmints Via d Gamd Optimulity Ctiteria Method, Ph.D.
Thesis, Deprtmsnt of Aeronautics and Astronautics, Stanford University (1976). II. M. W. Dews and R. B. Nelson, Application of optimality critk to automad stnlctural design. AM.4 I: 14(10),14361443(Oct. 1976). 12. L. Berke and N. S. Kbot, Use of @timabty Criteti Mathoak fz-
SE+ Systems.AGMD Lectw Series on Strucoptaaluhoa. PP. l-29. AUARIXS-70 (1974). 13. R A. GeRatly and L B&e, Optiaml structwd dcs&n. Tech. Rap. No. AFPDLTR-fO-l65. Wd&t-Pat&son A.F.B., Ohio
(1971). 14. R. A. Gdiatly, Developnwnt of procedures for large scale design. Tech. Rep. No. autofwed minimum w&&t strwtd
AFFDL-TRd6-!lUJ. Wri##-Pattetson
A.F.B., Ohio (1966). Appkbon of odmliwcdt&yrprooEbesto~Qairpforiargc
15. V. 8. Venkayya, N. S. Kbot nnd L. Me.
16. A. C. Ugural akd S. .K. l&&r, .i&nced St-h of Mat&ah Eiseviar, New York (1975). 17. A. K. Govil, J. S. Arora aad E. J. Haug, Substwturing in and a user’s manual for optimaldesi@lofwii#stwtuRs pmgr~&y .Tac/+ Rep. No. 36. p. 110. Mat&is Division. agwwmg. The University of Iowa. low8 City
(Mar. 1978).
Vol. IO.pp.m-910 Press W.. 1919. Prinled inGrealBrihin
OPTIMAL DESIGN OF WING STRUCTURES WITH SUBSTRUCTURINGt A. K. GOVIL,$ J. S. ARORAO and E. J. HAUL IbtcridsDivision, College of Et&wing The Universityof Iowa, Iowa City, IA 52242,U.S.A. (Receioe428 Auge~t 1978;rcceiwxf for publicutionI I January 1979)
into a stnxtd
opbibon
dgofithm.
1. -ON
recent paper[II, a generalmethodfor cplcdatins designderivatives of a.mcmaI mponsc quantities,such asstressesanddisp&ments,using&cconceptofsub sbuctWeswasdeveloped.Themethodutilizcsstate space of de* sen&ivity analysis in structural . . ideas . ptun~mn[2,3].Anewmethodofcakukt&design &i~ath with sub~bucturing wan inbgded into the gradient projection method for optimal structural design[4].Theln&hodwasapplkdto#eneralhussesand wasshowntpbemoreeBickntthanasimilarmethodU1 thatdoesnotuses&aWMn# Thepwpoacoft&ispaperis;Oextendapplkationsof theoptimalde&nmetbodwithsub&WW@tocompkxw-typestru&res.‘l%eopWaldes$nprobkmis formul&dinSectionxwheretb:colrtfunctionand f&re constraints for the wing des@ probkm an fk&Rd.Themainfeaturesoftheoj&naldesignaL In a
gkthlllwith~~Surmmaized.Detai)ed
derivationofthea&ithmisomittufintltkpaper,since it has been presented in a previous paper[4]. Extensive lrpplicrtioDsoftbeuWodanpresuWinSection3. Sevcaalnurn&alexa8r&sarecons&edandtheir optilllumsoWonsalecWsp8r&witlkotherresultsthat have been given in Refs.ET_ls].In order to have valid comp&ons,thcuniteelementnKMklsusedinI2efs. P-151 are also used in the present formulation to model the stnXtu@s. IWHMALaglcRaFwtNG~ TheoptiiDaldesignprobkmforidealiMwin$structures is de6ned as follows: IInd the design variabk assocktedwitheachekmentofawingstructure,such thatthetotalweightisatini&&,theequationsof e@&rium hold, and cons&a& on stress, dispkcement. andmembersixeansati&&Theclassoftwoand thret-dimensionalsWuctumlsystems considuzd herein is assumed to have a pteassioled geometric con&nation
and to be constructed of a known stn&tral material.
Furtheritisassumedthattbesestructurescanbeidealixcd using the following three types of finite ekments[lZ]: (i) truss ekmcnts of uniform cross-sectional arta (ii) constant strain Mngular (0 membrane elements of uniform thkkness, and (iii) symmetric shear panels (SW) or symmetric pure shear panels (SPSP). It is noted here that the SSP ckment diflers from the usual SPSP ekment in that it carries normal and shear loadsI6). Asanexampkofwingdesigqconsiderthestructure showninFii. l.Thestru&ueissynunetricwithrespect tothtx*-x~~one,soarlythcupperhplfofthestructme is modekd with CST, SSPISPSP, and ttuss ekments. Figure 2 shows the finite ekment i&&z&on of the stru&re. It consists of U nodes, 20 truss &men& (spar caps), 6ocsT ekments and 70 web eklmnts. The structurehasl2Odegreesoffreedornandl5fJliniteekments. whensparcapsarenotinchxkdinthefiniteekment ideal&ion, the structure has 130ekments. In analysis andoptimaldcsign,tkstructwemaybedividcdinto scveralsubstruc~s.sohltions forsevefalidealixations
ofthestructurearepresentedlaterintbepaper. Design variabks for wiqg structures are taken as cross-sec&al areas of the tn~s ekments and thicknesses of the CST and SSP/SPSP ekments. The design variabks for various types of ekments may also be combined into several groups to main&n symmetry in the structure and to reduce fabrication costs. The cost function, which is taken as the total weight of the structure, is an explicit function of the design variabks L
R(r)
Ilo(k)
NM(I)
(1)
where?‘P(r) = number of types of ekments in the rth subs&ucture;NGU) = number of groups in the &thtype of element; NM(i) = number of members in the ith group; b, = design variabk for ekmcnts of the ith group; &,= kngth of the jth truss ekment or surface area of meseerctl sqBportedby U.S. Army Anlmment Researcll and Devdapacat Commend, B&stics Research Labomtory, Aber- the jth CWSSPlSPSP ekment of the ith group; pc= weight density of ekments of the ith group; Z_. = total deen ProvingGrowk Wrylend. number of substructures. SW, Appiied l&clmks Depmtment,M.N.R. En* The CST, SSP and SPSP ekments are required to Cowe.fuwebad,lndie. 0Associate Frofessor. satisfy a stress criterion based on the Von Mises ~Professor. equivalent stress (also known as Maximum Energy of
A. K. GOVIL et al.
1
Fii. I. IS0 (130) Ekmcnt swept wing.
D&or&~ Theory(l61). According to this criterion, an eqtdvknt stns~ (u’) for CST w S!PWSP) &mm is givena6 ug=(o:,+u~-ollu~+3u,3
2
I/2 (2)
where 0; is the direct stress for truss elements or the
maximumvonMisesequ&kntstresscakula&dfrom eqn (2) for CST/!BP/W§P &ments and UP is an allowable stress. Displacolnent constlaints
at verious points of the
9lUCtUE~Wli~aS
where uij are the stress components at the point of in&test. The stress constraint for a typicai element of the wing structure is (3)
(4) where
zj is a calculated tipplacement and I,“ is, its allowablevalue. Furally, explicit bounds on design variabks are expressed as
(5) where bL and h” are lower and upper bounds on the ith design variable. The optimal desii formtdation for wing-type struc-
tures d&ed in eqns fl)-tS).Bts into the general formulation for strucarral opti&&n treated in Ref. [4]. Detailed derivation of the optimal design *aithm with substruchuingis presented there. Computptionolaspects of the algorithmare also discussed. Here, only the main features of the algorithmand pertinent expressions needs later are summarized: (1) The method is iterative and is based on the formula u=O,l,2..... b ("+1'=b'Y'+8b(Y';
(6)
Here Y is the iteration number, b”’ is a starting design and &“’ is a change in desii at the vth iteration. (2) The chan# in design 8b”’ is derived using KuhnTucker necessary conditions applied to a linearized prohlem and is given as a”‘= -*‘+a2
(7)
where T)is a step size and 66’ and sb* are given a~[41 (8) fig. 2. I50 (130) Ekmcnt swept wing element numbering.
6&Z= -W-‘/+2
(9)
Optimal designof
B = ATW-'A.
901
wiaa&uetnresWithsubstructuring
Optimaldesigns for four examples are obtained usitg the program of Ref. (17). TbeSe reSuIts are compamd withotherresultSavaikbkintheIikmtwe.Itshouldbe (11) noted that optimumd&M reporkd in Refs. (7-101ale
Here W is a positive &finite weighting matrix to be selected by the designer(6],A is a matrix whose columns are design derivatives of active constraints, and i is a vector containing values of active constraints at the current design point. The step size n is calculated based on a desired reduction r in the cost function at the start of the iterative process and is given as(2.4-61
(3) The de&n derivatives of active constraints are cakukted using the substructural data. These calculations proceed one substructure at a time. (4) Several members for diflerent substructures may beaSs@nedthesamedeGgnvariable. (5) All design variabk~ are allowed to vary in each deSigniteration. (6) The quantity
obtaincd~diirerentcomputersin~aentoperotioa
environments. Therefore comparisons of compu& times presented are only approximate.OptimumdeSigns withtbesubstNcturingmet&dureobtmneduSingan IBM 3tXMXH) compumr with doubk precision calculations. Optimaldesigns presented in Refs. (7,8] were obtai& using an IBM 36O-91compmer, which is roughly8timesfasterthanIBM3&6S@ivatecommunkation, IBM). Optimnl de+s preSented in Refs. (9,101were obtained using a CDC-7600compukr, which is approximately five times faster than the IBM MO 91(9, IO]. Exam& 1: 63 Member wing-carry-thmngh mnciun (Wcrs)
F~3&owsatrussidedix&nofthe63member wing-carry-tkoughstNcture(wCTs),nodalcoordirrptes (iinchesinparentheSeS),andthetwoloadingconditions.ThiSprobkmwrsorigi&yopti&edbyBerke and Khot(l21 and later by S&nit and MiuraI7,81and Dobbs and NeIson(ll]. The &ucture k modeled by 63 truss ekments and ham42 degas of freedom[l2]. . Intheoptimalde@nformuktionbys~ (J&‘l)2 = w6’=wtw) (13) Thestru&ueisdividedintn~~bypuris monitored. As the optimum is ppprosdled[Z], it is titk&gitatnodeS7-lO.The6int~cc&ktS of 31 member%:l-8,17-20,23-n, M-33,42&, 5& 59. required that pf?q2+0. Theremammg32mem&rSareintbeSu!oudSub&ucture. Tabk 1 gives member detMimm for the 1ArrucATKmANDcoy~~oFllDIIIuLf8 stnktuIe.subStNctm%ouehaSl8mtmiordqqcesof IntbisSectionthe~appli&ibtyandcomfreedomaudl2boundmyde#reeSoffreedmuSub put&nal etRckncy of the algo&hm for optimal Sulk- structum2haSl2inteSkrandl2boun&ydegnX!Sof tlaaldeSignwitllSttbS~iSdemonStrated,tk0Itph comparisonof the optimal des&ns for sevgil structures with known solutkns in the liiurep-15J. All Struck tures are divided into a number of subStructures for structural analysis and de&n SenSitivityanalysis. All calculations in the a@rithm proceed substntcture-wise. For detdled dkunuion of e@cientnumerkal impkmentation of the al&&m the reader is referred to Refs. 11,461. CONDITION 2 Acomputerprogmmbasedonthealgo&hmofRef.[4] and the formuktion preSented herein is developed in Ref. (171.The propom is automatic in the sense that deS@erneedStosupplydataaboutsubStructuresatKl starting values for the de&n variabks. The program continues execution until converpence is 0btaM or until a limit on the number of iterations is exceeded. The desipm has an option of performing a number of
“~tre~s-ratk’ design cycks at the start of the a4~o&hm[6].This is a poSitive feature of the program thatallowsthede&nertoobtainareasonabkdeGgnat the start of the a@ori&m[4].It must be noted however that even for the stress constrained design probkm the StreSS-ratiodeSi@cycles do not yield an optimumdesign for statica& i&terminate structures. When the structure is designed for multiple failure criteria, such as stress, displacement, bucllio& natural frequency, and OtherC0nStraintS, the stress-ratiodesigncycks produce a design that may be quite far from the optinuun. The authors have found that stress-ratio design cycles give a reasonabk initial design estimate calkd for in the Step 1 of optimal structural design algorithm with substructurinnl4.61. CAsVd. M?u.CE
FRONT l-30.0)
l-30.40)
VIEW
(-30.00)
(-30.120)
‘O& 12
TOPVIEW
+XI
IWE
I ) * COORDINATES IN INCHES
Fig. 3. 63 Member ~~&-carry-uuousb stmctuc.
A. K. GOVILerai. T&k 1.Memberlocationsfor63 membersWCTS
1
1.3
17
3.5
33
8.6
49
7.4
2
2.4
18
4.6
34
7.u
50
9.14
3
1.5
19
7.9
33
8.14
51
7.12
4
2.6
20
8.10
36
11.9
52
13.10
5
7.3
21
11.13
37
12.10
53
11.8
6
6.4
22
12.14
38
11.17
54
13.18
7
9.5
23
1.2
39
12.18
55
11.16
8
10.6
24
3.4
40
15.13
56
17.14
9
11.7
2S
5.6
41
16.14
57
15.12 5,4
10
12.8
26
7.8
42
1.6
58
11
13.9
27
9.10
43
1.4
59
3.6
12
14.10
28
11.12
44
5.2
60
9.8
3.2
61
7.10
13
lS,ll
29
13;14
45
14
16.12
30
3.9
46
5.10
62
13.12
15
17.13
31
4.10
47
3.6
63
11,14
16
18.14
32
7.S
48
9.6
Tabk 2.Results for 63 member WCTS I(rkr lhds 1
2 3 4 5 6 7 0 9 10 11 12 13 14 15 16 17 18 19 :: 23 24 25 26 27 28 29 30 31 32
37.4460 37.5610 52.7160 53.6230 23.8050 28.8300 17.1540 21.4020 26.2760 25.09so 8.8686 8.9586 23.4150 19.5970 5.1817 3.0207 37.0940 37.3540 0.0100 0.0100 0.0921
38.0780 36.1020 51.9230 54.2430 25.0690 28.2220 17.4610 20.7660 24.8100 26.9370 7.5854 9.0049 24.0470 20.4880 4.2966 2.9538 37.0600 37.1710 o.olOO 0.0100 0.0470 0.1048 1.1187 0.0312 2.9wO 1.3034 4.7730 0.8632 2.61109 2.7707 5.8034
0.0100 0.0100 0.0100 4.1985 0.9146 3.2270 0.0100 0.0127 8.0586 9.1997
Ud@t io lb NO.
I(u.
of
Activa
Colutrdnt
I
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 :: 54 55 56 57 58 59 60 61 62 63
optimum Deoipl in ln.L Robla
l(A)
5.6645 6.0951 5.6651 2.81S6 2.7665 2.6920 2.7716 5.7531 5.6651 16.0290 18.3970 11.4900 13.8640 11.73SO 6.1777 11.7930 13.9880 7.3SO8 7.7067 5.3699 0.021s 3.5992 9.7775 4.2407 0.1179 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 -4975.06
Constrdnt8
Violation
55 0.52 x lo-'
Problrl(B: 9.0427 9.6634 9.0430 8.4160 8.0605 7.9197 a.0632 9.264s 9.0417 24.3540 26.0850 19.3100 21.0440 17.0110 12.0530 18.4540 19.6470 6.5775 5.6535 0.4259 3.8917 6.1081 12.1060 5.7593 1.5657 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 -7 6117.38 23 0.34 x lo-=
3.68
0.91
21.64
39.b6
903
Optimaldesignof wingstructunswith&structuring Table3. Computationaltimes in seconds for 63 memberWCTS 1 Problm
Tote1 CPU
Tote1 CPU Time For Problem Setup
1
Streer-Ratio
1.987
l(B)
1.988
Design*
(2) + (3) + (4)
4
5
97.242
8.955 (lo)+ 11.641
Time
Opt-1
3
2
l(A)
Design
108.182
62.221
(13)+
75.850
*Inch&s CPU time requiredfor snidysis, sensitivity adysis and opfimhtion.
tNumbcrof stress-ratio designi&rationsperformed. freedom.Thus the entirestructwehas 12 boundary
degrees of freedom.
The structu~ is designed for two sets of constraints: (i) with stress constraints only (Problem l(A)) and (ii) with stress and torsional stillness constraints (FWbkm l(B)). Tbe torsional sti&ss constraint of Ref. [121is imposed by Emit& the relative displacementof nodes 1 and2intbex,dire&m.Nodesignvariablelinkingis used so there are 63 design variables for the structure. Design data for the WCT?Jare: material density, p f O.Mb/in’, modulus of elasticity (Titanium alloy) E= 16xlO‘psi,limitonreWivedispkcementofnodes1and 2 io the x~=direction=l.Oin., lower limit on crossse&mal areas - 0.01in’, upper limit on cross-sectional ar@a?i=noae, initial value of cro!Ss-sectionalaEas= 20.0ia’. Tabk 2 gives numerkal results for this exampk. For Probkm l(A), the set of active constmillts at the optimum in&&s stress in members I-15.17, 1%23-25,28, 5oaad5zuedWkadillgcoaditioo1,aadinmembers26, 30,32-3&3&49andu-56%underloadingcondition2. Minimmnsizeconatraintsonmembers19,2O,22and #)a3~eboective.ForproMem1(B),tbesttofadive . ums@wtsPttheoptimwnin&dcsstrcssinmcmhcrs 1-8,l&18,21,28,29,50 and 51, under loadingcondition
1,aadthetor&malstiffnessunderloadiqj0mdition2. Probkm l(A) rtquired 16 and P&km l(B) required 22 iteMionsoftheoptimaldesignaIgo&ntoconverge. The cost function historks are as follows: Robkm l(A): 4979.8, 4991.3, 4989.8, 4990.3, 4987.9, 4986.1, 4985.4, 4974.3, 4W.5, 4982.1, Ml.2 4980.6, 4979.2, 4977.9, 4976.7,4975.9,4975.1;and Problem l(B): 4977.5,5593.4,
6067.5, 7153.8, 6169.7, 6171.4, 6121.2, 6126.1, 6118.2. 6120.8, 611.7.9, 6139.4, 6147.1, 6144.7, 6144.2, 6138.9 6138.9,6134.2,6128.7,6124.0,6121.0.6118.5,6117.4. Wads of computa&maltimes with the substructurin8 algorithmare given in Table 3, sod Tabk 4 gives a comparison of results with other methods. The cost function is practicallythe same as repo&d in Ref. [7Jfor boththe probkms,whereasin compa&mwithRef.[12]it is about 1%lower.In compsr& withRef. [l 11,the tW weightachievedby the presentmethodis about1%lower for~l(A)and996~f~RoMcarltB).Finslly,it canbeseenfromTabk4thatthcreiscunderabk dll~iilcomputlaatiIllCUsiIlgtbC~dCsi$ll algwithmwithsUbs~
Example2: 18EIa~ntwingk~xbi~n
Fii4showsgeomebya1~ldime&msofw18 CkIWBtWiQ#bOXbC?ZMlWbdtWlOdii&coedi(ionr.TbG strwum issymmetricwidlItspect to fhe q-12 #me" Tbu&odytheapperlmlfoftbeboxbmmisitidw.
TWocasesfurthisexampkpfo&m8recxm&led In P&km 2(N), 5 busr, SC!JTand 8SSP dew&e wed.ForPr&km2@),SPSPelemm@musedinstd of SSP elements.In the optimaldesignformulntionwith subsbucturisgtheiddzedwingstnkctu~isdivided illtotwosub&uchWypprtitioaiapitrtnodes3and4. The firstsubstnsctrnecon&s of 2 trusselements(1.3). 2 CSTekments (1,2) and 3 SSPDPSPelements(1,3,6). the remainin ekments are in substructwe2. Both substructureshave6boundqdqrecsoffradom.Thefirst subsbucturchas no interiordegreesof freedom,whereas the second subsmlctufe has 9 inttxia degree!?of
Tabk4.Compn&onofrcsuksfor63membcrWCTS t
weight fn ibn
Probla l(A) Av. CPU Tke Per Cycle
Tote1 CPU T*
Prrrnt
4975.06 (la)*
6.08 (l.o)+
108.18 (1.0)'
Sckit-BKure [7] Jkrke-Khot [12]
4976.00 (14)* 5034.50 (50)*
6.43 (6.4)+ --
PO.47 (6.7)+ --_
Dobbe- Releon [ll]
5026.40 (ll)*
--
-_-
Problem l(B) Preeent
6117.4 (22)f
2.83 (l.O)+
75.85 (l.o)+
SctrPit-Miura [7] Berke-Kbot[12]
6120.9 (13)* 6159.3 (50)'
6.70 (113.9)~ -
87.51 (9.2)1 -
Dobbs-Nelson[111
6646.0 (20)*
ONamber of dssisncycks. tRelativecola& effort,
---
_-
I
A. K.
CST
c
SSP
et
al.
LOAD CONDITION 2 1DX103 Ibs
c> TRUSS 3
GOVIL
LOAD CONDITICN
1
Fig.4.I8Ekmcnt wingbox beam. freedom. Design data for this example are: p = 0.10lb/it?; E = 10X lb psi (Akminum alloy); Y= 0.30; 2,’ = 22.0 in. in hdhection; Op = 210 X lo’ psi; the bWerlimi~theupperhitandthestUtingareasfor ~NMekamnta are 0.02, I.0 and 0.98in’, respectively. For the remainingelements, tbe lower limit, the upper limit and tbe stating thicknessesare 0.02. 1.0 and 0.1%in., reSpcctivsly. Tabk 5 gives final results for both cases. The set of active con&mints at the optimum for Probkm 2(A) includee de&&on at node 5 in the x&ire&n under load&an&on I; Voa Miaesequivaknt stress for CST elansat 5 aad SSP ekments 3$ and 8 under loading coadith IandforSSPekments 1 and2underlonding co&ion 2; and mitt&m ekment size for truss ele-
ments 2-5 sod for SSP elements 6 and 7. For Probkm 2(B) the active comdrahts in&de ddktion at no& S in the x,-direction under load& an&ion 2; Van Mises equivakntstnssforCSTdement2iudSPSPd#nm:s 1 and 2 under loadingcondition 2; aod minimumelement size for truss ekments 2,3,5 and SPSP ekments 6 and 7. Problem 2(A) required 15and Probkm 2(B) requhd 9 opt&d design iterations to converge, The cost function histories are as follows: Probkm 2(A)do1.2, 424.4, 377.8.414.4,377.0,381.9,4fM.9,377.6,399.9,410.4,411).8, 406.4. 404.0, 403.0, 402.6, 402.7; and Probkm 2(B)355.9,359.7,354.1,360.8,360.7,360.2,3S8.9,358.8,357.!, 357.1. Details of computationaltimes with the sttbs&uc~
Tabk 5.Results for 18elementwingboxbeam
l-
WI 'robla 2(A)
3.o4.2
cst (k.1
3
s
6
1.2 3.4 5
0.004363 0.0s3070 0.037644
0.112730 0.053630 0.0364U
1 2 3 4
0.370soo 0.22#70 0.126540 0.120790 0.091634 0.02oaQO 0.02OWO 0.030954
0.086141 0.089aO 0.07462b 0.073004 0.073266 0.02OUM 0.02OOOO 0.024866
402.73
357.09
:
$sP For Prob. 2(A) kd
9 10 :;
6?sP lor Prob. 2(B)
13 14
:
tip.)
15 16
:
UeQbt ia lba. 8’ ii
lo.
t:iZ
O.low O.lom 0.1000 0.1000
of rtiva constraint
butraiat* vioktioll
0.10660 0. sbs22 0.1oooO
9 0.46xl63 44.16 l2os.00
Optimal design of wing structures with substructuring Table 6. Computationd times in seconds for 18element wing box beam TOM
cm
. Tiu
Totalcto
for
Dul#n*
w%+w
4
3
5
1
2
204)
0.847
l.SOS(4)+
9.800
12.152
20)
1.015
1.316(4)+
5.593
7.924 ,
*InChIdes CPU timereqnhdfor analysis, sensitivity anaiysis md optisniution. tNumber of stress-ratio design intcr&ions perform&
ofrcrubfor18ekmmtwingboxbcam T&k 'I.Coarporison Robla
’ uelgbt
Z(A)
Av.
Total Th
CPU The per cycle
ia lbr
CPU
Rewnt
402.7
(lS)*
0.65
(l.O$
12.15
(l.OF
Sdmit-Wure [71
403.0
(8)*
0.35
c4.3>+
3.37
(2.29
Itirsi [Q.lOI
402.7
(lS)*
0.06
(3.4)+
0.97
(3.2$
oa1lat1y
I141
_-
___
--
Cdlatly-Berke 1131
--
Z(B)
Roblar
Recent Scldt-Nlere171
357.1
(Q)*
0.62
(l.O)+
7.92
(1.09
357.8
(8)*
0.36
(4.6)+
3.43
(3.5)+
r9.101 cellecly-Berke (131
3a7.7
iurrl
w-e
___
(4)*
We_
-_
(193)*
___
-_
-em
389.8
cplletlY[141 *Number of de&a cycks.
tRektiWtzamphgeaort.
Tabk8.WLta
for 33&signvari&krectrnylprwi~
9brelldeerqndx2 llede
o=Penxt Iawl
L
of
mte
of bed ix
cod1thn
1
x3
lad
dirextioa
l
0.0
(kipd
bed bild:tial2
3 4 5 6 7 8 9
0.1656 0.4144 0.1426 0.3564 0.2311 0.3274 o.u77
0.2996 0.3745 0.2S76 0.322l 0.2367 0.2958 0.2127
10
0.2943
0.2659
11 a2 13 14
0.0978 0.2445 0.0381 0.0953
0.1768 0.2209 0.0689 0.0861
906
A. K. GOVIL eraL 0
TRUSS
14
Fig. 5.33&signvuiabk rtctmguhr wing.
Table 9.RCSU~~S for 33 design~a&&
E 21 22 23 24 BS? (in.)
:: 27 28 29 30 31 32 33
1 2 3 4 5 6 7 8 9 10 11 l2 13 14 1S
Ic g
W&&t lo. of hethe In lba.Comtrdnts Msx. Constraint Viohtion
:. -
11~~112 I16b1112
recw
0.08067S 0.059261 0.0414fP 0.064171 0.031335 0.047776 0.026171 0.03S431 0.0111733 0.028262 0.010 0.023174 0.010 0.010 0.010 100.344 10 o.S8xlo-7 102.70 17.46
wiry
0.0?))019 0.0727lb 0.041699 o.e6sm3 .0.029265 0.010s10 0.022464 0.04077s 0.014S46 0.031025 0.010 0.0131SS 0.010 0.010 0.010 100.212 15 o.'72atlo-7 102.70 1.94
901
Optimal designof wing structureswith substructuring
algorithmarc given in Table 6. Table 7 gives a summary cumpksun of results with uthcr methods. Compared to R&L [7,9], the cost function is practicaliy the same, whereas fur Prubkm 2(B) there is a reduction of abuut 9%, compared tu Rcfs. (13, IS]. The present method shows 0 computinptime advantage of u factor of two to fuur fur this relatively small scale problem. EuwJPfiw&; ~~~~~~
us
25x lo’psi. Fur truss elements biL= 0.010in2, hii0= 1.5in2.and the starting design is 2.0in2. Fur the rcmaining elements btL = 0.01in, br” was not spccihd and the starting d&n is OMin., except for CST ckunents 1-12 forwhichitisO.oBin.Thae~trwo~conditions forthestruc~whkharegiveninTabk8.Alsudesign variabk Iii is used for CST ekmcnts.
*
for a ~~~~.~t~&n~~T~9~~~u~s~,~hiss~~~~~~ xpf2 p&meand currcspunds to the wing middle surface. ~n~~~~l~~~~~~4~~n. Both designs arc usabk, since all cuj+nts are Tks,oniytheuppcrhaifoftherectmq&rwingis mud&d, usii@12 truss &ncuts to represent spar cap& satiskd. Also, the wci@ts at the twu rtembc~ arc 12 CST elements for the skin, and 15 SSP elements for practically the same. Huwzvcr, the final desipas arc somewhat d&rent and the vahms of m’b a~ quite the vceical webs. In m design formuMun with substructuriag,the different. The value d Ii%& 8t the 42nd ituatiun is muchsmauerthanattht15thiWatilm.Furtk.tbecost idc&ed*&ucturcisdividcdintothrccsubstructures by parti&& it at nodes 5-6 and 9-10. Truss func&misveryncaritsoptimumvalpeafWr15itm+ ekmcnts l-4, CST ckments Id and SSP ekncnts l-4 tions, but it takes 27 man iterations to signifk&y reduce the value of iWj2. This behaviur is oberved in aud13formslWtructurclandtnlsse~nts5-8,cST &mm&s 5-8 pad SSP elements 5-8 and 14 form sub ~ys~~~~~~~tf~~ &mar. S~Z~~~~~~~~S~~ ~~tof~~~at~~~~s 3,~~~~~~~~S~f~. ~~~~s~at~~4~~x~ ~~~~~rnf~~~~~~ under luad& cor&iun 2; the Van Mises eq&vaknt and third s&tr~ues are 6, 12 and 12, rcspetively. strcssinsSP&mcnts land3,undertomiil@cuxiMun Ikaign data for the probkm arc: p*O.lOlb/in’, E= 10.5X lobpsi (Ahtminumalloy), v T 0.30, zj’ = 211.0in. 2; and the minimumelement size fur truss elements 11 in the x3 dir&ion at nudes 13 and lrlf and 0: = and 12,CST ckmcnts 11and 12Mesignvariable 16).nnd
1.262
19
33
0.206
5%
z:
2.380 0.910
z 36
0.363 0.431
33 ii
2.013 2.393
z
0.302 0.144
1.764
26
:;
0.269 0.395
0.727 1.366
fz
1.297 1.906 0.570 1.196
z 3l 32
tlz 43 44
X:S 0.169 0.116
ii 3s
0.646 0.402 0.396 O.lS4 0.3ll 0.462
0.363
. 19
20 21
1.02s 0.3% 0.643
it
1.374 0.825
24 23
0.264 0.663
E s :
z
0.651 0.224 1.092 0.5l6
z
OAS
3:
O-SOB 0.175
": 44
i%i O.l33 O.pB6 0.231 0,566
A.
908
K. GOVIL et al.
SSP elements 11, 13. 14 and 15. The total computing time with the present algorithm, which includes initial set-up time (2.09sec). 4 iterations of stress-ratio design time (3.53 set), and 15 iterations of design time (17.69 set), is 23.31 sec. In Refs. [9, IO], the CPU times reported are 0.19 se&sign iteration and 3.14 for 15 design iterations. Comparing the CPU time per design iteration and total CPU time with Ref. [lo], it is observed that the substructuring formuiation yields a reduction of factors of 6.4 and 4.5, respectively. Lastly, the cost function history (in Ibs) for this problem is: 64.48, 8255, 99.79, 104.31, 104.61, 102.07. 101.10, 100.80, lOtl.M), 100.18, 100.12, 100.36, 100.36, 100.35, 100.36,100.34. The optimum cost function is pm&ally the same as in Refs. [9, lo]. It is noted that for this problem components of I%’ and 86’ associated with design variabks for ekments 11 and 12 (truss) and 31-33 (SSP) are equal to xero after the 4th iteration. Hence, in the subsequent iterathms these design variabks are kept fixed [6]. ErMlplr 4: 150 (130) dcrnat swept wing F-s 1 and 2 show geometry, dimensions and element numbering of the ideal&d swept wing structure. As noted ear&r, Only the upper half Of the swept wing is modekd,using2Otntssekmentstorqnesentthespar caps, 60 CST ekments to qresent the skin, and 70 SSP
elements to represent the vertical webs. The wing without spar caps is referred to as Probkm 4(A)and with spar caps (truss elements) as Probkm 4(B). in this example, extensive design variable linking is employed (see Table 11). The number of independent design variables describing the spar cap, skin and web are 14,ll and 7, respectively. For Probkm 4(A) there are eighteen designvariablesand for Probkm 4(B) there are 32. In the optimal design formuktion with substructuring, the idea&d swept wing structure is divided into two substnktures by partitioning at nodes 17-20. The first substMcwn consists of eight truss ekments (l-4,1 l-14), twenty four CST elements (l-24), and hvettty-flve SSP elements (l-4, 11-14, 21-24, 31-34, 41-49). The remaining ekments are in substructure two. Both substructures have 12 boundary degrees of freedom. The fkst has 36 interior degrees of freedom, whereas the second has 72. LIesii data for this example are: p = 0.0% lb/m’, E = 10.6x lo” psi (Ahuninium alloy); Y= 0.30, z,” = 60.0 in. in x&ire&on, and uia = 225 x ldpsi. For truss ekments 6,‘ = 0.01 in’, b,” = 1.50in2, and starting design is 0.02 in*, and for tbe remaining elements b,‘ = 0.02 in. and br” are not specifkd. For Probkm 4(a) the starting desii is 0.20 and 0.10 in. for CSI‘ ekments l-24 and 25-60 and 0.2Oin. for all SSP ekments, respectively. Thereare twoloedinpconditioasforthestructure,~given in Tabk 10.
T&k 1I. Results for 150 (130) element sweptwing
.1
1
3 4 ii :
1: ll
cm (to.)
12 l3 14 15 16 17 16 19 if 22 23 24 25 26 ii 29 ii! 32
: 4 5.6 7.8 9.10 ll 12 13 14 15.16 17.18 19.20 l-6 7-12 u-la 19-W 23-36 37-u m-60 l-4 3-10 11-14 13-20 21-24 r-z 3340 41-49 soda 59-70
0.203wo
0.177250 O.lwtso 0.1292% 0.11m10 0.094651 0.02oam 0.632631 0.02OOoO 0.033091 0.046713 0.222400 0.0979ao o.O9a411 0.010172 0.035310 0.061103 O.l6W 2462.34 13 o.*a~lo-~ 33.12 2647.00
O.OlOOO8 O.OlWW 0.01OoOo o.olOooo o.olmoo 0.01moo o.olO@6o 1.5OoOfM 0.637270 0.339730 0.716llO 1.0029Oo 0.369730 o.OlcmO O.lRZ(IO 0.171290 0.13174Q 0.123uo o.o99201 0.067910 0.02W 0.032648 O.o2oooo 0.03u125 0.017229 0.228710 0.11sa60 0.064393 o.os3397 0.033617 0.06a786 0.0993% 2442.% 22 0.w x104 35.11 3263.00
909
Table 11 gives numericsi results for both aiscs of this exampk.ThesetofactivecolIs~attBtoptimumfor Probkm YA) includes van Mises eqltivaknt stress for CsT&neats8,14and#)underl&ingconditkn1, s!@cIcments20,21,30,S8,and61u&rIoadilQumdition 1, and for SSP ekmellts 3, 11, and 42 under Ioadiq comMkn2,andlllhimumekment*forcsTekments 4%0 and SSP eIements5-10; Problem 4(B)was the same as for Frobksn 4(A) with addith of minimumarea for ~eiclneatsl-l0,19~2O,Pnd~~rveafor~ss eIemuIlt11. ~~~~~kf~~~~~~ in I&. f7, Ial, with the exception of the stress con-, saniatforSSPakmcdllamieri~conditioa2.Itis not#dthatnoncofthctipdelkcthconstraiatsart active. The d&but&~ of mat&l achiewd by the pMIltmCthodisappKhl@lythCMCaSiURCf.[71, cxccptforthcspnrcaps.TheoptimuJndesiimportcdin Tabk 11 is obtahl in 15 design iterations for both uurcs.ItisnotcdthatforRobkmya),a~desienis foundattbc1Othiterahwitbaweigbtof2469361b. wbicb is within 03% of the minimum achieved. For qtoMcm4tB)~ausabkdes@nisfoundatthe7thitera~~~~~~59.52lb, which is within 0.7% of
~_f~~~~~lS~~~~s~~~,~~~~~s~~f~: Probkm YA)-2340.1, 2432.7, 24715, 2460.1, 2459.8, 2457.3, 2452.7, 2467.4, 2463.2, 2469.4, 2467.0, 2465.4, 2444.2, 24fi3.2, 2462.7, 2462.3; and problem rySH440.8, U13.4, 2465.1, 2422.1, 2429.8, 2455.0, 2459.5, 24S7.0, 2454.1, 2448.7, 2443.9, 2443.4, 2442.7, w2.ig2442.5, 2442.1. DctaiIsof computationaitimes with the subst~cturing Table 13.Compa&onof
are given in T&k 12. Table 13 gives a compa&m of nsultswithotbcrmethods.Thccostflmctioa,ascompared to Rd. [9], is pm&ally the s8mc for botb cases. Itisthesrmefor~~YA),oscolnprradto~.Ttl, butabwt0.9%IessforPrdhmYB).Comqtingtimcby thepreseatmetbodisagainafactorofhvotoanordeiof m@tudelowertlumothermethods. Lastly, the swept wing is divhl into five sub&wtures by parthnii it at nodes 9-12.17~20.25-28 and 33-M. Nodes 41-44 are also umsidcred as boundwy ~S*W~~~,~~~~~~~~~~~~S*
~~nrn~~~~~~for~ ~~~A)~~).~~~PU~is~~~ same.TbisisdueotothcfactthattberotainPRIberof caIcuiptions with two ami five substructures is approximatelythe same.
sensithity analysis[2,3] and by incorporrdrm of subS~~~~~~.~U~cal-s~~htbcpnperarc~t inthcdesignoflwgcscakS-.Basedoat&
nsultsofthisnsearcb,itisconclu&dthatagencral purpoaopthUddC&llcomputerCodeWitbslIbstructlEin8 may be developed for a large class of strwtmkl design probkms,
fesldts for 150 (130) &men~ Swept wise
f
Problr
I -ight
in
Pr-t Schrit-Miura atrzi
[ 7]
I9.101
lbe
2462.3
(lS)*
2462.8 2461.8
Sclmit-Hiura Rlzri
r9.101
[71
I Tot&l CPU Time
4.61
(l.O)+
82.12
(LO)+
(8)*
2.53
(4.4)+
21.50
(2.1))
(16)*
0.75
(6.5)+
13.19
(6.4)+
ProblPresent
4(A)
CPU Nu per cycle
AV.
4(a)
2442.1
(U)*
4.94
(Lo)+
88.37
(l.o)+
2460.8
(9)*
3.69
(6.0)+
34.65
(3.1)+
2445.0
(16)*
1.16
(9.4)+
20.21
(9.2)+
‘Number of desii cycks. tR&tive computing &Tort.
910
A. K.
GOVIL et al.
1. I. S. Arora and A. K. God. Design sensitivityanalysiswith substructure. 1. &ng Mech. JXu. Proc. of ASCE. 103 (EM4), 537-548 (Aug. 1977). 2. E. J. Haug. Jr.. K. C. Paa and T. D. Streeta, A computathlalmatlwJdforoptimalrtrucanldasign:!piecewise uniform structures. ht. I. Namer. M&h. Engng 5, 171-184 (1972). 3. E. J. u and J. S. Arora, De+ sensitivity analysis of elastic iwcbicd systems. Coarpmter Methods in Applied Mechanics and EN&W& Vol. IS, pp. 35-82 (19%). 4. J. S. Arora and A; K. God. an etlkkt method fur optimal structural design by substruenrrinl. comput. StIwcttf#s 7(4). 507-515 (1977). 5. J. S. Arota and E. J. Haug, Jr., EtIkicnt optimal design of structuresby ganaahd swpcst &swat pmlramming. ht. /. Nunw Met. .Qtgng lU4), 747-766 (1976); and IU6), 14XL1426 (1916). 6. A. K. God, J. S. Arora aml E. 1. Haug, Substrwturintl
meulodsfnrdesign~aadysisandsmictmalopti&don. T~~.No.34(&oPh.D.Diswwbnofthe firsta&or)* kbt&ab Dildoas, cone@of E?&eIriqt, TbaulBksityofIowsIowaCity,lewr(Aug.1ppT). 7. L. A. S&nit, Jr. and H. t&n, &edm&~ Cvts for E.#ciant Skctuui Syntksk NASA CR-2552, University of
Califarnia, Los A&es, CA %I024(Mar. 1976). 8. LA.SabmitaodH.ffiAnew~ad~sisbn.
on optimatity criteria. Pm. of AIAA/ASME/SAE 17th s)NCtures, Stmctund @namics and Material Conf., pp.
448-462. King of Prussia, Pennsylvania (5-7 May 1976).
10.P. Rid, Optinizcltionof Structures with Complu Constmints Via d Gamd Optimulity Ctiteria Method, Ph.D.
Thesis, Deprtmsnt of Aeronautics and Astronautics, Stanford University (1976). II. M. W. Dews and R. B. Nelson, Application of optimality critk to automad stnlctural design. AM.4 I: 14(10),14361443(Oct. 1976). 12. L. Berke and N. S. Kbot, Use of @timabty Criteti Mathoak fz-
SE+ Systems.AGMD Lectw Series on Strucoptaaluhoa. PP. l-29. AUARIXS-70 (1974). 13. R A. GeRatly and L B&e, Optiaml structwd dcs&n. Tech. Rap. No. AFPDLTR-fO-l65. Wd&t-Pat&son A.F.B., Ohio
(1971). 14. R. A. Gdiatly, Developnwnt of procedures for large scale design. Tech. Rep. No. autofwed minimum w&&t strwtd
AFFDL-TRd6-!lUJ. Wri##-Pattetson
A.F.B., Ohio (1966). Appkbon of odmliwcdt&yrprooEbesto~Qairpforiargc
15. V. 8. Venkayya, N. S. Kbot nnd L. Me.
16. A. C. Ugural akd S. .K. l&&r, .i&nced St-h of Mat&ah Eiseviar, New York (1975). 17. A. K. Govil, J. S. Arora aad E. J. Haug, Substwturing in and a user’s manual for optimaldesi@lofwii#stwtuRs pmgr~&y .Tac/+ Rep. No. 36. p. 110. Mat&is Division. agwwmg. The University of Iowa. low8 City
(Mar. 1978).