SuTdung phUcrng phap phan tiirhuru han tmh toi Uu kfch thirdc dan
Size optimization of truss using finite element method
Ngay nhan bai: 19/3/2017 Ngaysufa bai: 5/4/2017 Ngay chap nhan dang: 5/5/2017
Vu Thj Bfch Quyen, Cao Quoc Khanh
TOMTAT
Bli bao trinh biy mot thu?t toan mdi tinh toi ilu kich thilcic <ikn phang siJ (iung phUdng phap phan tCf hiJu han.
Tren ccf sd do thi^t lap chiicing trinh tinh toi liu kich thiicic dan phang biing phSn mem Matlab
TiJ khoa: Toi ilu kich thdde dan
ABSTRACT
This paper presents new algorithm of truss sizing optimizing problem using FEM. Hence, develop compiling program for calculating of truss sizing optimization using Matlab Key words: Sizing optimization of truss, FEM
TS. Vu Thi Bich Quyen Giang vien, Khoa Xay dUng, TrUdng DH Kien tnic Ha Noi
KS. Cao Quoc Khanh Can bp ky thuat
1 . Bat van de
Bai toan toi uu ti6a ket cau la bai toan tim kiem giai phap c6 loi nhat trong qua trinh ttnh toan thiet k^.TCrnhieu thap ky qua toi Uu hoa da tr6 thanh m6t phan khflng t h ^ thieu trong qua trinh thi^t ke so bo he ket cau. NgUdi thi^t ke can tim ra mot phuang an vi kich thu6c, hinh dang, v3t lieu (bien thiet ke) 3i dat duoc muc tieu nhd nhat ve trong iUOng hoac gi^ thanh mS v i n dam bao cac yeu cau ve do bin, ciJng va on flinh. Phu thu6c vao bien thiet ke bai toan tdi Uu dan dUoc dUOc chia thanh ba loai. tdi Uu kich thudc; toi Uu h5nh dang, toi Uu cau true. Khi thiet ke dan, tdi Uu kich thUdc la bai toan thUfing gap trong qua trinh thiet ke so bp, vdi bien thiet ke dudc chon la kich thudc mat cat ngang, toa dd ciia c a c n u t v a l i ^ n k ^ t lacodmh.
Ly thuyet thiet ke tdi Uu 1^ mot trong nhCing hiidng nghien cUu phdt trien nhanh trong linh vUcco hoc vat ran bien dang va co ket cau, la sU ket hap giUa ly thuyet ca hoc va ly thuyet toi Uu. Viec gicii de bai toan tdi Uu dUoc thuc hien tren co s6 Ung dung cac phuong phap toan hoc. Khdi dau viec giSi cac bSi to^n tdi Uu duac thuc hi§n bang each sUdung dc phuong phap giSi tfch. Tuy nhi§n, phuang phap giSi tfch thudng gap phai cac khd khan ve mat toan hgc, chi co the ^p dung doi vdi cic ket cau don g i l n . PhUOng phap giSi tich gan nhukhdng cdkhS nang g i l l quyet nhCfng bai toan thucte vdi dieu kien bien va tell trong philc tap.
Su phat tn^n manh me cCia cdng ngh§ thdng tin va cac phuang phap sd, dac fai&t I I phuang phap phan tU hQu han, la mgt cong cu hUu hi^u trong viec g i l i de bai t o l n ket cau, trong dd cd bai toan tdi Uu ket cau dan.
Trong b l i bao trinh bay dUdng ldi giSi bai t o l n tim ki^m nghi&m tdi Uu kfch thudc dan b i n g mdt thuat t o l n lap mdi do t i c gia de xuat. Cac dieu kien rang budc ciia b l i t o l n tdi uu duac thiet lap tren c o s d phuong p h i p phan tUhUu han.TUdd viet chuang trinh tinh tdi Uu kich thudc dan phang b i n g phan mem l i p trinh Matlab.
2. Bai toan t o ! Uu ket du
Bli t o l n tdi uu ket cau cd dang chung bao gdm: cac bien thiet ke, h i m muc tieu v l dieu kien rang budc.
H a m m u c t i ^ u Z = F(X)- (max) Vdl cic bien thiet ke ' ^ - { ' ' i
^ . . ^ , g , ( X ) ( < ^ ) b , ; i = l - r He rang buoc
(2.1) (2.2) (2.3) Trong c i c phUOng phap g i l i bai toan tdi Uu, phuang phap quy hoach toan hoc la mdt cdng cu kinh dien, tdng q u i t va cd hieu luc. Trong bai t o l n quy hoach toan hoc, he r i n g budc md ti vec to bien X n i m trong mien r i n g budc D thudc khdng gian thUc n chilu.
g,{X)(<->)b,;i = l - m ) X e D , c R " J
Phuang phap quy hoach t o l n hoc la phuong p h i p tim nghiem tdi Uu trong n c i c h xuat phat tU mdt diem Xo ban dau
tim hudng den diem Xi,Xj,Xi,...bSng each l i p d i n Xp
(2 4) n thiet ke D bSng
0 5 . 2 0 1 7 iSOIIillKllSX 9 3
Xp-{xl'",xf„.,x;^' xl,"'} (2.6) khi ham muc tieu khdng the nhd hole Idn han dUoc nUa ma v l n
thda man c i c d i l u ki^n rang buoc.
3. Tinh ndi IUc va chuyen vj dan phSng b i n g phUcfng phap p h a n ti]fhuiu han
Trinh t u g i l i bai toan d i n p h I n g b i n g phUOng p h i p phan tif hUu han [91 gdm cac bUdc chinh nhU sau:
- Rdi rac hda dan thanh cac phan tU, thiet l i p ma tran chuyen vi, ma t r l n t l i trong, ma t r l n dd cUng ciia he,
- T h i l t lap phuang trinh g i l i trong he toa dd chung cd ke den dieu ki^n bien. Gili he phUOng trinh x i c dmh chuyin vt tai niit p h i n tit trong he tga do chung:
y ^ K - ' . P (3.1) Vdi: K la ma tran dd cUng ciia he d i n ; y I I vec to chuyen vi nut ciia
he; P la vec t o t l i trong tai niit dan.
- Xac dmh ndi lUcvaiJngsuattrong phan t l i d i n
-t ->,
Vdi' E la md dun d i n hdi; A I I dien tich m l t c i t ngang; I la chieu d l i phan tif; TE la ma tran chuyen true ciia phan tif; 6, I I vec t o chuyin vi nut phan tU trong he toa dd chung
4 . Thiet lap bai toan t d i Uu dan p h ^ n g theo phUcmg phdp phSn tufhQuhan
4 . 1 . Ham muc t i l u
Trong b l i toan tdi uu kich thudc d i n ham muc t i l u dUoc chon I I cUc tieu hda trgng lUOng G cua t o l n bd he dan.
G = Z T . ' - , A , (4.1)
Trong d d :
• N I I sd nhdm bien t h i l t k l . Biln t h i l t k l I I dien tfch m l t c i t ngang A, d e t h u a n t i e n t r o n g c h l t a o , k e t c a u dan duae chia thanh mdt sd nhdm cau kien cd dien tich mat c i t ngang nhU nhau, chilu dai cd t h i khlc nhau
• Ll, A I : tdng chilu dai va dien tich meit dt ngang cOa cac phan tU
• Vl' trong iuong riSng ciia v i t lieu nhdm i.
Trong trudng hop k i t cau ddng c h i t thi ham muc tieu cd dang the tfch k i t cau
(4.2) 4.2. S i l u kien rdng buoc
a Rang budc ve dd ben:
4^1
(4 3)
Trong do: [ a ] ; [ a ] ^ I I Ung s u i t cho ph^p khi k^o va k h i n l n ddi vdi phan t i n
Rang budc v l dd eUng; yp < [ y ] (4.4) Trong dd: [y] la chuyen vi cho ph^p vdi cac d i l m d i t t l i trong
Rang budc ve c i n b i n g thda man dieu kien phuang trinh g i l i bai t o l n ciia phuang p h i p phan tU huu han
P=Ky (4 5) 5. Phutfng ph<ip lap t i n h t o i Uu t r p n g lUtfng dan p h ^ n g
De tim nghi&m tdi Uu trong luong dan phang cho bai t o l n da thilt lap dtren t i e g i l thiet lap mdt phuong phSp lap.Trinhtutinh dUocbSt dau tU diem ed the tich d i n Idn nhSt trong trUdng hop cae thanh dan cd ciing tiet dien, Trong q u i trinh tinh tdi Uu t h i tfch t i c g ! l de xuat mdt h? sd va dat ten la "he sd tUong quan ndi luc giUa cac thanh d i n ' H I sd n l y t h i hien quan he ndi lUc glUa cic thanh d i n , dUOc sirdung lam ca s i cho viec lUa chon lai tiet d i f n cho cac thanh trong q u i trinh l i p . Trinh t u g i l i b l i toan tdi Uu trong luong dan bao gdm c i c budc ca b i n sau.
BUdc 1;Xlc djnh so bd kieh thUdc m l t cat ngang.
G i l dinh elc thanh cd cung kfch thUdc mat c3t ngang A.
SU dung phuang p h i p p h i n t i i hUu han tinh chuyin viy, ting suit a. ndi lUe f eiia eac thanh. Cac h i m e h u y i n vi. Ung suit va ndi luc cd the b i l u dien phu thudc vao ^n sd (bien thiet ke) A dudj dang sau:
y.^a,.A"'; Oj = b,.A''; f =0,
Vdi a, b, c I I cac h i n g sd dUoc xac dinh tU phuong trinh g i l i bing phuong phap phan tUhUu han.
Xle dinh A^m tU dieu kien ben v l chuyen vi cac p h i n tU
^ : ^ = > A ^ j , = m a x j ^ ; j ^
Xac dinh the tich d i n so bd V, = ^ L , A ,
BUdc 2:Xlc dinh lai kfch thudc m l t c i t ngang theo h i sd tuang quan ndi lUc.
Xic dinh he sd tUflng quan ndi luc ciia mdi thanh (h? sd nly the hien ty le giifa ndi luc giijfa c l e thanh). Ve n g u y i n t i c cd the ehon ndi lUc ciia mdt thanh bat ky lam gia tn chuan fi Tuy nhien khdng nen chon nhifng thanh cd t i l t dien tiem can g i l tri khdng de tranh sai sd. Xlcdinh duae he sd tUOng quan ndi lUc
' ^ f, _ C,
n. Y~Y
Tren cd sd h? sd tUOng quan ndi luc chgn iai t i l t di§n elc thanh theo cdng thUc A,=niAi
ThUc hien trinh t u t l n h n h u bUde 1 vdi cac g i l tn Ai vUa duac chon lai tai budc hai.
Xle dmh t h i tich d i n Vj = ^ L , A , Xac dmh % g i l m t i l t d i l n
1 = ' ^100%
Thuc hien cac bUdc lap nhU t r l n d i n vdng thU n cd p h i n tram gilm tiet dien dat g i l tri theo yeu c l u .
Budc 3:
In k i t qua tinh va ket thuc
6. T h i l t lap chUdng t r i n t i t i n h tdi Uu t r o n g lUOng d ^ n ph&ng sC d u n g phan mem IVIatlab
Tren cO s6 ly t h u y l t tfnh ndi lUe c h u y i n vi d i n phling bSng phUOng phap phan tU hUu han va phuong p h i p l i p d l de xuat d tr^n, t i c g i l d l t h i l t l i p s d d d khdi ( h i n h l ) v l Viet chuong trinh tfnh tdi Uu dan phling bang phan m i m Matlab [2].
94 eninniiiBii 05.2017
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1
;,r.r;:rf?;rr.i^;r
i
..<[.] U-N'IHJ"C KfeT THtiC J
Hinh 1. S(f dokhoi nit gon diifong trinli tinh t5i uu dan phJng sil'dung phuong phap phan ti> hull han
7. Vi du tinh t o l n
Sil dung chuong trinh tinh tdi uu dan phang da t h i l t l i p [2j, t i c g i l da thUc hien cac v i d u tfnh tdi Uu d i n phling cho cac loaf d i n khac nhau D l k i l m chUng va so sanh ket qua tinh vdi cac phuong p h i p khac, tac g i l se trinh b i y vi du tinh tdi Uu dan phSng tronghlnh 2. CIc sd l i l u tinh toan nhu sau:
Trong luong rieng; p = 2 . 7 7 x l O - ' ( N / r a m ^ ) IWd dun dan hdi vat lieu. E = 6 9 0 0 0 ( N / m m ^ ) Qng suit cho p h i p : [o] = ± 1 7 2 ( N / m m ^ ) Chuyin vj cho p h i p : d - ± 5 0 . 8 ( m m ) T l i trgng: P „ = P,, =445000(TM)
Quy luat thay ddi dien tieh mat elt ngang elc thanh theo thU t u vdng lap the hien tren do thj tren hinh 3 Cd the nhan thay sau l l n 4 ket qua tinh elc thanh d l u hdi t u . Tren hinh 4 the hien do thi bien thien h i m muc tieu tdng the tich dan qua elc vdng lap. Sir dung thuat toan lap do tac gia de xuat ham muc tieu ludn hdi tu vdi tdc do tuong doi nhanh.
hi bien thien dien tich mat cat ngang cac thanh d^n theo thiiti/vong lap TdNG THf nCHD.4.\(iii
Hmh 4.06 thi bien thien the tich dar theo thUtUvong lap 8. So sdnh ket qud tinh v d i cac phifcmg khac khlc 8.1. So sanh k i t qud tfnh vdi p h i n m i m HyperWord [11]
Phin mem HyperVi/ork la mdt phan mem dUOc v i l t tren co sd phuang phap phan tU huu han. HyperVi'ork eho phep thuc hien cle p h i n tich tuyen tinh v l phi tuyen k i t cau k i t hop vdi k h i nang toi Uu hda t h i l t ke. Trong nghien cdu [11] da md phdng dan phang bSng phuang phap phan tU hUu han, sau dd thue h i l n tinh tdi uu d i n b i n g phuong p h i p lap. Cle k i t q u i tinh tdi Uu dan khi sU dung HyperWork v l c h u o n g trinh tinh do tac gia thiet lap duoc the hien tren do thi hinh 4. Phan tich quy i u l t bien thien ciia c i c dd thj nhan thay ket q u i tinh tdi uu trong luong d i n theo chUOng trinh tae gia da t h i l t lap hdi tu nhanh hOn, cd g i l tri nhd hon so vdi k i t q u i tfnh b i n g p h i n mem HyperWork.
Hinh 2 So do dan phJng chiu tSI trong linh
SV" THAY BOI TRQNG LroO G DAN
43 1 — -
• 3 2,9 0 %•>
^;ss» '
^ ^^*^^^*
^1
ry
2 3.30
3 1.1s
4 J.0D
5 ' 6 1 7 2.85 2.JS 2.J3 8 2.72 i.n
»
i.n 10Hinh 4 fio thi bien thien tiong iuong dan theo thU tir v6ng lap 8.2. So s i n h ket q u i t i n h v d i cac nghien cihi cija cac t i c giii khac Trong nhUng nam g i n d l y , b l i t o l n tdi Uu trong lUOng d i n tren da duoc nhieu t i c g i l t r l n t h i gidi thi/c hien bSng cac phuong phap khlc nhau nhu phUOng p h i p thuat giai di truyen [6,7,8,10], phuong phap luc ket hop thuat toan tim kiem [3,5], phUong p h i p tinh t o l n tdi Uu h o i dua tren thuat t o l n Metaheuristic [4]. Cac k i t q u i tinh duoc the hien trong b i n g 1.
Barig 1. K i t q u i tinh tdi Uu trong lugng d i n theo cic phuong p h i p STT
b
1 2 3 4 5 6 7 8 9 10
W
(lb)
!31
33,50
1,62
22,90
13,90
1,62
1,62
7,97
22,90
22,00
1,62
5479
oid^HUATdTM&uiemmimEockH&iitNa^ (ii^)
[4]
30,00
1,62
22,00
22,00
1,62
1,62
7,97
22,00
22,00
1,62
S567 [S]
33,50
1,62
22.90
14,20
1,62
1,62
n , 5 0
22,00
19,90
1,62
5517 [6}
30,52
0,10
23.20
15,22
0,10
0,55
7,47
21,03
21,53
0,10
5061 17]
29,35
0,10
23,79
15,11
0,10
0,51
7,37
21,04
22,21
0,10
5064 [8!
30,87
0,10
23,47
14,51
0,10
0,53
7,50
21,17
21,01
0,10
5070 [10]
30,51
0,10
23,20
15,19
0,10
0,56
7,46
21,07
21,47
0,10
5058 TAC Gii
28,90
0,0J
28,35
14,44
0,00
0,01
B,24
20,43
20,43
0,02
S065
Id dc so lieu trong b i n g 1 cd the thay k i t q u i tinh tdi Uu trpng IU(?ng d i n do tac gia thUc hien c d s u chenh l§ch khdng d i n g k l (0,1%) so vdi cac ket q u i tinh b i n g phudng phap t h u i t giai di truyen. t h i p hOn d i n g k l so vdi cac ket q u i tinh b i n g cac phuang phap cdn lai (den 10%). Oac biet phuong p h i p tmh do t i c gia thiet l i p cho k i t q u i vdi sd vdng l i p can thiet thap hon rat nhieu so vdi cac phuong p h i p khac
9. Nhan xet
Trong bai b i o t i e g i l d l de xuat va k i l m chdng mdt thuat t o l n mdi g i l l bai toan tdi Uu kich thUde dan p h I n g t h i l t l i p t r l n co sd phUOng p h i p p h i n tU hiiu han. Trong cac nghien cUu t i l p theo t i e g i l se tilp tue phat trien t h u i t t o l n da xay dUng d l g i l i bai toan tdi uu cho ket c l u dan cd k l den yeu t d phi t u y l n hinh hqc.
TAI LlEU THAM KHAO
[11 Le Xuan Huynh, Tinh todn kel cdu Iheo ly thuyet toi uu, Nha xuat ban khoa hoc va hy thuat, 2006.
[2] Cao Qudc Khanh, Nghiin cuu phuong phip tinh tdi mi dan phdng. Luan van thac si ky thuat x l y dung cdng trinh d i n dung v l cong nghiep, TrUiingfiai hoc Kien triicH^ MDI, 2017
[i\hitiawiteiOm\aYitaim^osseini,AhybridflS-CSSalgorithmforsHnullaneousanolysii, designondoplimiiationoftrussesviaforcemethad,PenoitaPoiytedinicaCimilitqtiteermqS6ll
mi).
[4] All Kaveh & Ali Zoighadr, A mulli-sei charged system search for truss optimwiion iwtft variables of different natures; element grouping. Periodica Poiytechnica Civil Engineering (S5/2 2011).
[5] A.Kaveh and M.Hassani, Simlllaneoui analysis, design and optimization alstrucBim usin3ftrcemefftod(m(*!!nfco/ony(j/gijr;fhmi,As!a|Ournal ofcivi!engineering vol 10,no4(2009|.
[61BehroozFafshi and All Alinia-ziazi, Sizing o;)lin)izanDna/rf(i«strucrureiJ'ynietMa/
wnfenrndfoireftrmufafion, Intemational journal of solids and struGures 47 (2010) [7] Dominique Chamoret, Kepeng (Uu and Mathlhieu Domaszewski, Optimization oflmss structures by 0 stochastic method, ASMDO (2009)
[8] Haitham Farah Hanna Al Rabadi, Tnisi sAf and topology optimization using hamtwi i e o f c f t m e f W , University of Iowa (2014).
[9] J N Reddy, An Introduction ta the Finite Element Method Third Edition, Teiias ASM University College Station, Texas, USA 77843 (2006)
[10] Reza Najian A;l & Mohamad As(ani & Masoud Shariat Panabi, Sizing Optimizahoit o(
frussStnicturesusingaHybndizedGenetKAIgoriSim.OuneJ.lOM)
[^^]S.S.V^i<ieSlK^iKai3m{lO]5),l)lscreteSl^eOptlmi^ationoflndelermlnateTltlssUsi^g
%fflVo*.JoumalofCivilEngineeringandEnvironmentalTechnology,(Volume2, Number 10)
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