M \l c tieu : giai thi�u mct phuang phap PTHH tran k@t hqp vai ky thu�t rai rc;icl�ch truqt va sau d6 chung toi ap dvng vao tinh toan phan tich mct s6 bai toan baitoan finh, dao dcng tµ
Trang 2Ml)C Ll)C
Ml)C Ll)C
DANH Ml)C HINH VE
DANH Ml)C BANG BIEU
MUCLUC• •
DANH Ml)C cAc CHU VIET TAT
THONG TIN KET QUA NGHIEN CUU
INFORMATION ON RESEARCH RESUTLS
Ill
V
Vll
Vlll
IX
CHUONG 1 TONG QUAN : 1
1.1 Gioi thi�u chung 1
1.2 Nhi�1n Vl,l d€ tai 2
CHUONG 2 Cd SO LY THUYET 3
2.1 Ly thuy�t co v�t rim bi�n d�ng 3
2.2 V�t li�u composite nhi€u lop _ 5 ,,
2.2.1 Lop lamina gia cucmg c6t sgi m9t phuong 5
2.2.2 Tfi1n composite lop :'! 7
2.3 Ly thuy�t tfim dan h6i Reissner - Mindlin : 8
2.3 1 Quan h� bi�n _!f�ng - chuy�n vi � 9
2.3 2 Quan h� (mg sufit - bi�n d�ng 10
2.3.3 N<)i lµc trong tfim 10
2.4 Phuong trinh nang lugng 12
CHUONG 3 PHUONG PHAP PHAN TU HUU HAN 14
3 1 Phuong phap PTHH truy€n th6ng 14
3.2 Ph�n tir rai r�c l�ch trugt (Discrete Shear Gap - DSG) 16
3.3 Phuong p'hap ph�n tir hfru h�n tron dµa tren nut (NS-DSG3) 21
CHUONG 4 ':CAC vi Dl) s6 25
4.1 Bai toan tinh 25
4.1.1 Tfim d�ng huong - 25
4.1.1.1 Khc'w sat sµ h<)i tv cua phuong phap 25
4.1.1.2 Khc'w saUmh huong cua ti 1� a/h 28
4.1.2 Tfim composite nhi€u lop 30
Trang 3Ml)C Ll)C
4.1.2.1 T�m sandwich 3 lap chiu tai phan bf> dSu ··· 30
4.1.2.2 T�m composite 4 lap [0°190°190°10°] chiu tai hinh si n va tai phan bf> dSu 31 4.2 Phan tfch tfrn sf> dao d(mg t\l' do 33
4.2.1 Tii1n di'ing huang 33
4.2.2 T'11n co1nposite lap 34
4.2.2.1 T'11n vuong , 34
4.2.2.2 Tii1n xien • 39
4.2.2.3� Tiiin·tron , 40 �
4.2.2.4 Ta1n tam gi ac 42
4 3 B' t ' ;: d ht a1 oan on !n a1n co1npos1 e ;_ ·1 1 · ap 44
4.3.1 Tiim chiu nen m<)t phuang 45
4.3.2 Tiim chju nen hai phuang 48
CHUONG 5 KET LUAN VA KIEN NGHJ 49
5.1 K�t lu�n 49
5.2 · Huang phat trien _ 50
TAI LI]$U THAM KHAO 51
11
Trang 5DANH Ml)C HINH VE
Hinh 4.10 Hinh d�mg va lu&i chia cho tftm trcm 41 Hinh 4.11 Sau ,d�ng dao d(mg d'1u tien cua t�m tron composite [ 45° I -45° I -45° I 45°], dieu ki�n bien ngam quanh chu vi 42 Hinh 4.12 Hinh d�ng va lu&i chia cho tftm tam giac 43 Hinh 4.13 Sau d�ng dao d(mg d'1u tien cua tftm tam giac composite [0°190°10°], dieu ki�n bien ngam quanh chu vi ···i···· �··· 44 Hinh 4.14 Tftm composite chiu tai nen: (a) m9t phuang; (b) hai phuang 45 Hinh 4.15 Anh hm'mg ti I� modun dan h6i Ei/E2 dSn tai t&i h�n 46 Hinh 4.16 Hinh d�ng cua tftm composite l&p khi chiu tai t&i h�n (mg v&i cac dieu ki�n bien khac nhau ··· 4 7
IV
Trang 6DANH MQC BANG BIEU
Bang 4.16 T1n s6 dao d(mg tµ nhien cua tflm tam giac ngam theo chu vi v�t lieu M3, kich thuac a/h = 100 43
Bang 4.17 Tai tai h?n cho tflm composite [0°/90°/90°!0°] tµa dun tuung (mg v6i ti I� E1/E2 thay d6i 45 Bang 4.18 Tai t6i h?n cho tflm composite [0°/90°] va [0°!90°!90°10°] tµa dun tuung (mg v6i ti I� kich thuac a/h thay d6i 46 Bang 4.19 Tai t6i h?n cho tflm composite 2 16-p [0°/90°] tuung (mg v6i cac di�u ki�n bien khac nhau ··· 4 7 Bang 4.20 Tai t6i h?n cho tflm composite [0°/90°] va [0°!90°/90°!0°] tµa dun tuung (mg v6i ti I� kich thuac a/h thay d6i 48
.' -
vi
Trang 7DANH Ml)C BANG BIEU
DANH MUC BANG BIEU
Bang 4.1 Thu(k tinh v�t li�u cho m6i lap lamina trong tftm composite lap 25 Bang 4.2 :89 vong w = IO Ow Eh 3 I ( 12(1- v 2 )qa 4
) tc;ti tam tftm 26 Bang 4.3 Momen M=lOOM I qa 2tc;ti tam tftm 26 Bang 4.4 :89 vong va thanh phcln ung suftt cua tftm sandwic�- Srinivas 31 Bang 4.5 :89 vong t<;ti tam tftm 4 lap d6i xung g6c SQ'i cheo [0°190°190°10°] chiu tai tr9ng hinh sin 32 Bang 4.6 :89 vong tc;ti tam tfrm 4 lop d6i xung g6c SQ'i cheo [0°190°190°10°] chiu tai phan b6 deu 33 Bang 4 7 Sau tftn s6 dao d9ng tv nhien m cua tfrm vuong bien tµa 34 Bang 4.8 Sau tftris? dao d9ng tv nhien m cua tftm vuong 4 C<;tnh ngam 34 Bang 4.9 Sau tcln s6 da9-d9ng W nhien m cua tfrm vuong [0°/90°10°] (a/b=l) ngam 4 C<;i,Ph tuang Ung vai ti 1� cc;tnh/chieu day a/h.' 35 Bang 4.10 Sau tftn s6 dao d9ng tv nhien m cua tftm hinh chfr nh�t [0°190°10°] (a/b=2) ngam 4 cc;tnh tUO'ng ung VO'i ti l� C<;tnh/chieu day b/h 36 Bang 4.11 Anh huang ti l� dai/chieu day (a/h) len tcln s6 dao d9ng tv nhien cua tfrm composite 4 lap [0°/90°/90°/0°] bien tva don 38 Bang 4.12 Anh huang ti 1� modun dan h6i Young Ei/E2 len tfrn s6 dao d9ng tv nhien w = m( b2
/ 1r2 )� ph/ D0 cua tfrm composite 4 lap [0°!90°!90°!o0 ] 39 Bang 4.13 Quan h� gifra tcln s6 dao d9ng W nhien cua tftm xien composite 5 lap [90°10°190°10°!90°] va g6c xien a , 40 Bang 4.14 Quan h� gifra tcln s6 dao d9ng tv nhien cua tfim xien composite 5 lap [ 450/_45o/45o/_45o/45o] Va g6c xien a 40 Bang 4.15 Tfrn s6 dao d9ng tlJ nhien cua tfrm tron 4 lap [8°;-e0;-e0;e0] tuang
ung va1 goc x1en cot SQ'l
V
Trang 9THONG TIN KET QUA NGHIEN c(J'u
THONG TIN KET QUA NGHIEN CUU
1 Thong tin chung :
- Tend� tai : Phuong phap phfin tu huu hc;in tran va (mg d\mg vao phan tich
k@t c.lu t.im
- Ma s6: T2010-04
- Chu nhi�m : Trfin Trung Dung
- Dan vi cua chu nhi�m d� tai : Khoa Xay Dµng va Di�n -:
Thai gian thµc hi�n : tu 04/2010 d@n 04/2011
t-2 M \l c tieu : giai thi�u mc)t phuang phap PTHH tran k@t hqp vai ky thu�t rai rc;icl�ch truqt va sau d6 chung toi ap dvng vao tinh toan phan tich mc)t s6 bai toan baitoan finh, dao dc)ng tµ do va 6n djnh k@t c.iu t.im dfing huang va t.im composite
3 Tinh m6'i va sang t�o : d6 la sµ (mg dvng phuang phap phfin tu huu h,;m tran nut
k@t hgp ky thu�t rai rc;tc l�ch truqt vao phan tich k@t c.iu t.im composite lap
4 K�t qua nghien CU'U : K@t qua nghien cuu cho th.ly SlJ hi�u qua cua phuang phapphfin tu hfru hc;tn tran nut ap dvng cho lap bai toan t.im composite lap khi so sanhvai cac nghien cuu da dugc cong b6 truac d6
5 San ph§m : cod� Matlab tinh toan va 4 bai bao dang tren cac tc;ip chi qu6c t€,
trong nuac va hc)i nghi Cl) th€ nhu sau
TT Ten tac gia, Tun bai vi�t, nam xu�t ban Ten t�p chi, h(li nghi i
,,
Chien H Thai, Loe V Tran, Dung T Tran, Applied Mathematical
T.Nguyen-Thoi, H.Nguyen-Xuan, Analysis of Modelling
smoothed discrete shear gap method,
doi: 10.1 Ol 6/j.apm.2012.01.003
2 Tran Trung Dung, dz.mg phuang phdp PTHH tran nut cha bai todn Thai Hoang Chien, Ung T�p chi khoa h9c Trnong DH Mo TPHCM
Tran Trung Dung, Thai Hoang Chien, An T�p chi khoa h9c Trnong DH
Dung T Tran Chien H Thai, T.Nguyen-Thoi, Ki yeu hc)i nghi Khoa h9c toan
4 ,/ H.Nguyen-Xuan, A node-based smoothed finite quf>c Ca H9c VRBD l�n 10,
6 Hi�u qua, phu·O"ng thfrc chuy€n giao k�t qua nghien CU'U va kha nang ap d \l ng:
c6 kha nang ap dl)ng cao trong lap bai toan lien quan d@n k@t c.iu t.im, di;ic bi�t la t.imcomposite lap
Ngay 17 thang 01 nam 2011
Chu nhi�m di tai
vm
Trang 10INFORMATION UN l{�:S�AKLtt K!!,�U 1 L.:,
INFORMATION ON RESEARCH RESUTLS
1 General information :
- Project title: A smoothed finite element method (S-FEM) and an
application to analysis of plate structures
- Code number : T20 I 0-04
- Coordinator : Tran Trung Dung
- Implementing institution: Faculty of Construction and Electricity
4 Research results : The research results demonstrated that the present element isfree of shear locking and achieves high reliability and accuracy compared to otherpublished solutions in the literature
5 Products : Matlab code for composite plate problems and four papers published
on International Peer-Reviewed Journals in ISi (with SCI), National Journals andNational Conferences
No Authors, Title, Year Journal name, Conferences
Chien H Thai, Loe V Tran, Dung T Tran, ,'Applied Mathematical
T.Nguyen-Thoi, _H.Nguyen-Xuan, Analysis of MDdelling
shear deformation theory and node-based Impact Factor: 1.371
smoothed discrete shear gap method,
doi:10.1 OJ 6/j.apm.2012.01.003
Tran Trung Dung, Thai Hoang Chien, A node- Journal of Science Ho Chi
discrete shear gap for free vibration analysis of
laminated composite plates, 2011.
Tran Trung Dung, Thai Hoang Chien, An Journal of Science Ho Chi applicatj_pn of the node-based smoothed finite Minh City Open University
laminated composite plates, 2011.
Dung T Tran Chien H Thai, T.Nguyen-Thoi, Proceedings of the
4 H.Nguyen-Xuan, A node-based smoothed finite International Conference on
6 Effects, transfer alternatives of reserach results and applicability : The researchresults can very promising to provide a simple and effective tool for analysis ofcomposite plate structures
lX
Trang 11CHU'ONG 1: TONG QUAN
CHU.ONG 1
TONG QUAN
1.1 Gio·i thi�u chung
Ngay nay, phuang phap phfin tu hfru hc,1n (PTHH) dugc bi€t d€n la phuang phap tinh toan va mo phong s6 r(mg rai nhfit trong Khoa h9c - Ky thu�t Di€n hinh, PTHH la cong cv rfit hi�u qua trong mo phong s6 Ca - Ky thu�t, Ca luu chfit, Ca -Sinh h9c, Ca lugng tu, Ca - Di�n tu, Ca h9c v�t li�u, Coih9c Nano, V�t li�u ap di�n va cac bai toan v6i moi trucmg da v�t ly,
Tuy nhien, PTHH vfrn con t6n tc,1i nhfrng hc,1n ch€ lien quan d€n ky thu�t phfin tu: khi giai quy€t cac bai toan ca h9c Di€u d6 da anh huang dang k€ d€n d9 chinh xac va hi�u qua tinh toan cua phuang phap s6 thong dvng nay
Nhfrng hc,1n ch€ lien quan d€n ky thu�t phfin tu cua PTHH da va dang dugc giai quy€t trong va ngoai nu6c trong su6t thai gian qua Cac nghien cuu lien quan d€n vfin d€ nay da dugc cong b6 tren cac tc,1p chi qu6c t€ dfiu nganh (vi dv, International Journal of Numerical Methods in Engineering, Computer Methods in Applied Mechani�s and Engineering, Finite element and Design, Computational Mechanics, ) Hi�n nay c6 cac nghien cuu trong va ngoai nu6c v€ ky thu�t phfin
tu va vi v�y cac nghien cuu d€ tim ra phuang phap hi�u qua han vfrn khong bao gia
dung Nim bit tinh thail'sµ qu6c t€ v€ cai ti€n cac ky thu�t tinh toan PTHH d€ ung dvng trong Khoa h9c-Ky thu�t va xu th€ h<)i nh�p cua Khoa h9c -Ky thu�t trong nu6c, ch(lng toi gi6i thi�u m<)t giai phap cai ti€n hi�u qua tinh toan trong PTHH nh�m nang cao ki€n thuc h9c thu�t cua chung toi khi giai quy€t cac bai toan v6i muc de) phuc tc,1p ngay cang tang trong thµc t€, d6 la phuang phap phfin tu hfru hc,1n
tran (SFEM - Smoothed Finite Element Method) Day la phuang phap do Gui Rong Liu va Nguyen Thoi Trung d€ xufit, dµa tren k€t hgp ky thu�t m€m h6a bi€n dc,1ng [ 4] vao phuang phap phfrn tu hfru hc,1n truy€n th6ng SFEM xay dµng v6i 4 hu6ng ti€p c�n chinh: ti€p c�n dµa tren phfin tu (CS-FEM) [5], ti€p c�n dµa tren nut (NS-FEM) [6], ti€p c�n dµa tren cc,1nh (ES-FEM) [8] va ti€p c�n dµa tren m�t (FSFEM) [ 1 O] Trong s6 d6, phuang phap NS - FEM (node-based smoothed finite element method) dµa vao y tuang xay dµng mi€n tran la mi€n bao quanh nut phfin
tu c6 nhfrng uu di€m sau: cho k€t qua chinh xac han, va t6c d9 h9i t\l cao han cac phuang phap phfin tu hfru hc,1n thong thuang v6i cimg s6 lugng nut khao sat; khong cfin phai xay dµng ham dc,1ng m9t each chinh xac; ap dvng dugc cho ca phfin tu tam giac, tu giac va da giac n cc,1nh
Ben cc,1nh d6, hi�n nay v�t li�u composite 16p dugc t6 hgp tu cac 16p sgi va phuang cua sgi khong nhfit thi€t phai gi6ng nhau, chi€u day cua cac 16p thanh phfrn
trong k€t cfiu tfim composite nhi€u 16p thuang la nhfrng 16p mong, g9i la lamina,
dugc sip x€p theo tr�t tµ se thu dugc nhfrng tinh chfit nhu mong mu6n V�t li�u
composite nhi€u 16p (laminated composite) c6 nhfrng uu di€m nhu: cuang d9 cao,
de) cung 16n, tr9ng lugng nht;\ ch6ng an man.- mai man, tinh chfit nhi�t, gi6i hc,1n moi V6i nhfrng uu di€m nhu v�y, tfim lam b�ng v�t li�u composite 16p dang dugc
1
Trang 13CHU'ONG 2: CO so· LY THUYET
CHUONG 2
2.1 Ly thuy�t co· v�t din bi�n d�ng
V�t li�u di hu6ng la v�t li�u c6 tinh ch�t dan h6i khac n�u v� nhfrng hu6ng b�t ki, quan h� (mg su�t va bi�n dc,tng: a ij = c ij k/&kl .,
Hinh 2.1 Phep bi�n d6i tn.ic tc,m d9
Ma tr(in chuy.ln dbi: a,i =r � � � l
Tuong tv ta c6:
(2)
3
Trang 15CHU'ONG 2: CO SOL Y THUYET
2.2 V�t li�u composite nhi�u Io·p
2.2.1 L6'p lamina gia cll'c r ng c6t SQ'i m<)t phll'O'ng
Nghien cuu (mg xu ca h9c cua lop lamina trong k€t cfiu tfim composite, ta gia djnh:
• Tfim lamina la v�t li�u lien t1.1c, tuc la khong c6 16 h6ng ho�c t6n t�ikhoang tr6ng
• TfimJamina ung xu dan h6i tuy€n tinh-•
Ung xu cua v�t li�u tuan theo dinh lu�t Hooke
trong d6: 9 !i : ma tr�n h� s6 v�t li�u trong h� t9a d9 dia phuang (7)
Cac hfing s6 ky thu�t cua v�t li�u lien t1.1c composite gia cuong c6t sqi duqc xac dinh dµa tren nhfrng gia dinh sau [11]:
• Lien k€t gifra c6t gia cuong va v�t li�u nen la lien k€t ben
• C6t gia cuong phan b6 song song ho�c phan b6 deu trong k€t du tfim
• V�t li�u nen d6ng nhfit, khong c6 v€t nut va a tr�ng thai khong ungsufit ban dftu
• C6t gia cm'mg va v�t li�u nen deu d�ng hu6'ng va tuan theo djnh lu�t
• Tai tr9ng tac d1.1ng song song ho�c vuong g6c phuang cua c6t gia cuong.H� tfl_lc v�t li�u chinh cua k€t cfiu tfim composite gia cuong sqi m9t phuang bao g6m: m9t tr1.1c theo phuang cua sqi hay phuang ca ban, hai tfl_lc kia vuong g6c voi nhau v� vuong g6c voi tfl_lC tren, t�o thanh m9t tam di�n thu�n Mo-dun dan h6i young
va h� s6 Poisson cua v�t li�u composite nhieu lap duqc tinh toan dµa tren mo-dun dan h6i Young, h� s6 Poisson va kh6i luqng th� tich cua c6t gia cuong va v�t li�u nen:
5
Trang 20CHU'ONG 2: CO SOL Y THUY.ET
trong d6: µ = 5 I 6 : h� s6 di�u chinh c�t (shear correction factor)
Thay phuang trinh ( 19) va (20) vao phuang trinh (21 ), quan h� gifra cac thanh ph�n i'.mg su�t t6ng va bi�n d�ng cua lap lamina vi�t du6i d�ng ma tr�n nhu sau:
v6i
l A 11 A12 A
" l
A= A12 A22 A26 ,
A16 A26 A66
(22) (23) (24)
(25)
Trang d6 cac ma tr�n d9 ct'rng mang A , ma t�n d9 ct'rng k�t hqp B, ma tr�n
d9 ct'rng u6n D v6i (i,j) = 1,2,6 duqc dinh nghia nhu sau:
Trang 27CHUONG 3: PHU UNG PHAP PHAN TU HU U H�l� l KUl� 1�;.-,-u;.-,u-'
trong d6 ma tr�n d9 cung hinh h9c
v6i cac gia tri Bf va -r duqc xac djnh:
luqng bi�n d�ng u6n va kh6ng bi tri?t tieu khi chiSu day tfrm t � O, d§n d�n hi?n
tuqng "shear locking" lam lai giai kh6ng h('>i t\l va kh6ng con chinh xac f)€ giai quy�t hi?n tuqng "shear locking" c6 th€ dung tich phan giam (reduced integration)
cho ma tr�n d9 d9 cung cftt Ngoai ra, c6 th€ su d\lng ph�n tu b�c cao nhung trong m9t s6 truang hqp lai giai vfrn kh6ng h('>i t\l t6t va t6n nhiSu chi phi tinh toan
0€ gia quy�t vfrn dS nay, tac gia Bletzinger (2000) [2] da dS xufrt phuang phap "rai r�c l?ch truqt" - Discrete shear gap, xfrp xi thanh ph�n bi�n d�ng dt
thong qua gia tri m6i d6 la L1wr v6i m\lc dich giam b�c ma tr�n tinh bi�n d�ng cftt B5
xu6ng m9t b�c - tra thanh h�ng cho ph�n tu tam giac
)"
Hinh 3.2 Thanh ph�n l?ch truqt (shear gap)
16
Trang 29CHU'ONG 3: PHU'ONG PHJ\P PHAN Tlf HlJ'U H�N TRON NS-DSG3
Th€ cac gia tri <;, 17 t�i cac nut i = 1 3 phuang trinh (58) vao phuang trinh
(64) va (65), thanh phfin l�ch trugt rai r�c dugc tinh nhu sau
Liw<1l yl = Liw<3l yl = Liw<1l y2 = Liw<2l y2 = 0
Liw<2l = (w<2l _ w<ll) + !a(/J<1i + p<2l) + _!_b(/J<1i + /J<2i) y] 2 X X
2 J' J'
Liw;� = ( w<3l -w<1i)+ � d(/J;ll + fJ;3l)+ � c(/J;1i + /J;3l)
(66)
Thay cac gia td rai r�c l�ch trugt t�i cac nut phfin tu ( 66) vao phuang trinh
( 54) suy ra bi€n d�ng cit theo phuang <;:
·i-y,, , � = � oNi Liw; at r' = oN2 Liwat 2 r' = (w2 -w1) + _!_a(/J; + /J;)+ _!_b(f],1, + /J,:) 2 2 / (67)
va bi€n d�ng cit theo phuang 17:
Bi€n d�ng cit dang tinh theo t9a d9 dia phuang (<;,77), khi quy d6i ra t9a d9
t6ng the (x,y) ta dug�: bi€n d�ng dt theo phuang X -
06.W,; oi; oLiw,, OTJ oi; OTJ
Yx = ar- §x + fjry ox = Y,; ox + r,, ox
B�o ham cac chuyen vi th5ng theo bi€n x
auo "aN, I "(aN, a; aNi 817) I 1 [ 1 2 3]
U , =-= L -u = L + !su =- (b-c)u + cu -bu