Phan tich tmh hoc cua tam FGM diFa tren phtpcng phap khong \u>m va ly thuyet do'n gian biln dang cat bac nhat Static analysis of FGM plates based on the meshless method and simple first-order shear deformation theory
Ngay nhan bai; 02/12/2015 Ngay SLfS bai: 13/01/2016 Ngay chap nhan dang; 02/03/2016
TOMTAT
Bai bao nky gicii thieu mot mo hinh so mcfi phan tich tinh hoc ciia tam vat lieu bien doi chile nang vcii cac thuoc tinh vat li^u thay doi theo chieu day tfai, M6 hinh nay dila tren phifdng phap khong lil6i sii dung hara noi suy Moving Kriging (MK) ket hcfp v6i ly thuyet bien dang cat bac nh^t ddn gian (S-FSD) Cac vi du so dildc tiiilc hien de so sanh ket qua dat dUdc vdi cdc ket qua c6a cac nghiSn cilu da c6ng bo n h i m kiim chiing sii chinh xac cua rao hinh phan tich difOc de xuat.
Tii khoa; Tam vat lieu chiJc nang, ly thuyet bien dang cat bac nhat ddn gian, noi suy Moving Kriging, phildng phap khSng lu6i, p h ^ tich tinh.
ABSTRACT
This paper presents a new numerical model for analysing the static problem of the functionally graded material (FGM) plates in which material properties vary through the thickness. This model employed the the mesh-free method with Moving Kriging (MK) interpolation with the simple first-order shear deformation(S-FSD) theory. Several verification numerical examples are solved and compared with the other available numerical methods showing the accuracy of the proposed method.
Keywords: Functionaly graded plates. Simple first-order shear deformation theory. Moving Kriging interpolation, mesh-free method. Static analysis.
TS. Vii Tan Van, Giang vien, Khoa ky thuat Xay dilng, TrilOng Dai Hpc Kien True Tp.HCM
Email: [email protected]. vn Dien thoai: 4-84 123 686 9610 KS. Nguyin Bao Vinh
Hpc vien cao hpc, Khoa ky thuat Xay dilng, TnlcJng Dai Hoc Bach Khoa - Dai Hpc Qu6c Gia Tp.HCM
Email: [email protected] Dien thoai. +84 932 528 386
Vu TSn Van, Nguyen Bao Vinh
1 . Gidi t h i e u
vat lieu bien Soi chiJc nang (Functionally Graded IWaterial- FGM) 1^
mot loai composite c6 dac tinh vat lieu bien doi lien tuc trong vht the do 36 se loai bfi 3Udc hien tilpng tap trung ilng su^t thudng gap ci lo^i composite thong thitdng. FGM thu'dng dUOc che tao til hon hdp gSm gom va kim loai. Ohy la ioai vat lieu dang hiSdng nhifng khong dong nhat. Hien nay, FGM dtfoc quan t a m vi co the tao ra nhu'ng k^t cau cd, khh nang thfch ung vcii nhu'ng dieu kien v i n hanh. Thong thudng, phan tich ling xCr cila tam san xu^t tCf vat li§u chile nang (tam FGIVl) SiJpc dJa, tren b6i cac ly thuyet co ban sau: (i) Tam co dien (CP), (ii) Bien dang d t bac nhat (FSD), (iri) Bien dang cat bac cao (HSD).
Ly thuydt CP [1] khong xet den anh hUdng cOa bi^n dang cat ngang d^n ling x d cua t a m m o n g . Khi chieu day t a m tang l§n, bi^n dang cat ngang co anh hUcfng dang ke den dap ilng cua t^m Lf thuyet FSD [2-3] xet den anh huSng bien dang cat nay bang cacK xSy dung tru'dng chuyen vi tuyen tfnh bac n h ^ t trong mat phang doc theo chi^u day cila tam. Tuy vay, cac p h u o n g trinh can bang, on dinh dilHc xSy dyng dUa tren ly thuyet CPT va FSDT deu khong thoa man dieU kien bi#n ve sU triet tieu ilng suat 6 mat tren va du6i cOa tam. Nham gicii quyet dUoc kho khan nay, m o t he so dieu chinh bi^n dang cat duoc sil dung d e d i e u chinh m o i quan he ket hop gida Ung suat c^tva bien dang cat ngang. Gia trj he so dieu chinh nay phu thuflc vao cac thong so nhU hlnh hoc, tai trong tac d u n g , dieu kien bien cua t^m. lj thuyet HSD [4-21] xet den anh hUdng bien dang cat ngang bang each:
xay dung cac trUdng chuyen vi bac cao d trong mat phang doc theo chieu day cila tam, hoac theo mat p h i n g ngang cila tam. Cac phUcng.
trinh can bang, on dinh dUa tren trUdng c h u y i n Vj da thda man cac tat ca dieu kien bien. Tuy vay, viec phan tich ilng XLT cila t 3 m dUa tren cac ly thuyet HSD nay rat philc tap do so lUpng bien so d cac phuong trinh can bang, on dinh tang ien, chang han ham chuyen vi duoc xay diln|, tren ly thuyet HSD de xuat bdi Pradyumna va Bandyopadhyay [11], Neves va cong sU [13,20-21] s i l dung 9 an so; Reddy [9],Talha va Singh [ 1 2 ] s i l d u n g lan lUot gom 11,13 an so.
Dil cho mdt so ly thuyet HSD khae sil dung ham chuyen vi gom 5 ^n so tUdng tU nhU ly thuyet FSD chang han nhU. ly t h u y l t bien dang cSt bac ba (TSD) [4], ly thuyet bien dang cat ham sin [14], l i ' thuyet bien dang cat ham lUpng giac [15-17]. Tuy vay, phUOng trinh can bang, on dinh dat dUpc tU cac l>? thuyet nay van philc tap hpn so vdi ly thuyet bien dang cat bac nhat (FSD).
Ly thuyet b i l n dang cat bac nhat d o n gian (S-FSD) dUpc de xuat dau tien bdi Huffington [22] vdi ham chuyen v| chi g o m 4 in so.
Khae vdi I5? t h u y e t FSD, thanh p h ^ n goc xoay dUpc b i l u dien thong qua thanh phan uon va cSt tao nen trUdng ehuyen vi t r o n g mat phang, chuyen vl ngang cCia t a m .
H/lat khae, khi khao sat Ung x i l mat on dinh cua t a m FGM ehiu tac dung ciJa tai trpng phan bo phi tuyen trong mat phang tai cac canh bien ciia tam, Chen va cong sU [23] cung khang dinh rang phUOng phap
112
khdng ludi-sCf dung tru'dng chuydn vi xay d i l n g dUa tren tpa d p cCia cac nilt rdi rac trong cSu t m c se t r i n h du'pc nhiJng su phijfc tap ve so khi sCr dung cac loai phan t d trong phUOng phap phan tCt hOu han.
Gu [24] gidi thieu dang thilc mdi cua phUOng phap khdng lUdi dUa tr§n dang yeu Galerkin ket hpp vdi ham npi suy IVloving Knging (MK) gpi la philcfiig phap IWKG. (Wot trong nhCfng uu diem ciJa ham noi suy MK la thda man tinh chat cOa ham delta Knonecker, khic phuc dupe nhung t r d ngai ve d i l u Itien bien trpng yeu xhy ra ddi vdi phuang phap khdng lUdi.
Npi dung bai bao d l xuat m o hinh phan tich tmh cila tam FGM dUa veio l>? thuydt S-FSD ket hop vdi phuong phap MKG. IV16 hlnh vat lieu chile nang dupe trinh hky d muc 2. Ly thuyet ddn gihn bien dang e3t hke nhat duoc trinh bay d muc 3.
Mo hinh phan tich dUpc de xuat d muc 4. Vi du so dUpc thUc hien de ki^m chiing d d tin cay eua m o hlnh dupc trinh bay d muc 5. Sau cung la c^c k i t luan t h u dUcfc tCr md hlnh dupe nghign cilu n^u trgn.
2 . TKmFGM
Xet mdt tam FGM dUoc c h l tao t d vat iiSu kim loai va g o m co chieu dky h. Mat dudi va t r ^ n cda tSm hoan toan la kim loai va g o m . Mat phSng xy nam d gida t a m . Chieu ddOng cda true z hudng iin tren. Trong hhi bao nay, t J so Posslon's v ddpc xem 1^ hang so. NgUpc lai, mddun dan hoi E, mkt do khdi lupng p dupe xem la thay doi lien tuc theo c h i l u d&y tam FGM vdi luat hon hop Voigt hay theo IdOc do Mon-Tanaka [4].
Theo d d , m o d u n 3kn hoi E(z), mat do khdi lupng p(z)ddpcxac dinh nhUsau:
E ( z ) = E , + ( E , - E , ) V , (1) p(z) = p „ + { p , - p , ) V , (2) Trong dd chl sd m va c d^i dien cho thanh phan kim loai va g d m
tuong d h g ; V j = 0,5 + ham m u , the hien sd gia tang ty le theo chieu dky -0.5h < z S 0.5h .
la t h i tich thanh phan g d m ; n la chi sd cda phan the tich; z la bien tpa do
D :
' / / '
y ' 1 0 , ,
n =
y'
y i ^ ^
.-- ),,-'
1 = 3.
• '
i i -
.
— —
^
n = y
JL^
. ' • • '
1 - ^
• n =
_.
_ •
0 5 . ' n =
••-"
. ' •
D.3^
n .y'
/ /
= QA_
rttj^i
/'
1
%
• • / /
/ / / / /
3. Ly thuyet bien dang cat bac 1 dtfn giSn
Odi vdi ly thuyet b i l n dang cat bac nhat FSD [2-3J, trUdng ehuyen vj cua t a m (U|,Uj,U3) c d t h e ddpc bieu dien ddi vdi 5 bien so nhdsau;
u,(x.y,z) = u ( x , y ) - z 5 w , { x , y ) / a < (3) U2(x,y,z) = v ( x , y ) - z 9 w , ( x , y ) / 5 y (4)
U3(x,y,z) = w(x,y) (5) Trong dd u(x,y),v{x,y),w{x,y) la nhdng an sd chuyen vj cua mat gida
cua tam theo cac p h d d n g x , y , z t d P n g dng; 9,(x,y),<p,(x,y) la che gde xoay cda phap tuyen cda mat phang gida tam theo true x , y . Ly thuyet bien dang c3t bac nhat dPn gian [S-FSD) sil d u n g c i c gia thuyet sau d l lam don gian iy thuyet b i l n dang c3t bac nhat (FSD): (i) c h u y i n vj theo phdOng ddng gdm thanh phan c h u y i n vj do u d n w ^ v a catw^ gSy ra, nghia la: w[x,y) = Wb(x,y) +Wj(x,y) ,{ii) thanh phan gde xoay chi do thanh phan chuyen vj do udn gay ra:(p,(x,y) = - e w ^ ( x , y ) / 3 x (Pj,(x,y) = -5W|,(x,y)/3y ;.Vi vay cac cdng thilc (3), (4) va (5) cd the viet lai nhusau:
u,(x,y,z} = u{x,y)+Z(p,(x,y) (6) Uj[x,y,z) = v{x,y)+zip^(x,y) (?) U3{x,y,z) = Wi,(x.y) + Wj(x,y) (8) Khdng gidng vdi 1^ t h u y l t FSD, trUdng chuyen vi dupe xac dinh
theo cdng thde (6)-(8) chl gdm 4 an sd: u[x,y),v{x,y), w^(x,y) va w j x , y ) . Bdl vi thanh phan gde xoay la dao hhm bae nhat cda thJnh phan chuyen vi do udn tUong thich vdi sd rdi rac cua ly t h u y l t bien dang c i t bac nhat don gihn (S-FSD) tranh ddpc hien tupng khda c i t (shear locking).
Dda tren gia t h i l t b i l n dang nhd, mdi quan he gida b i l n dang va ehuyen vi ddpc bieu dien nhu sau:
9u 9 v _ a V i , Sy 9x 9x5y
Cdng thde (9) viet dUdi dang ma tran nhusg
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V^
HinhLQuanhegiDa V^ vat^ 16chieudSyz/h theochlsfiji.
Hinh 1 b l l u dien sd thay doi cua t h i tich thanh phgn gdm V^ ddi vdi t ^ so chieu deiy t ^ m FGM khi trj sd n thay ddi. Doi vdi gia tri n rat Idn n>100 thi VJ rat be - ed t h i xem n h u vat lieu cua tam ehi bao gdm la kim loaL Ddi vdi g i ^ tri n r^t b^ n < 0.01 thi V^ = 1 - cd the xem n h u vat lieu cua tarn chi bao g o m la g d m . Sd thay ddi cda viec k i t hop gida hai vat lieu kim loai v^ gdm la tuyen tinh khi n = l .
[ flu
ax
8v fly d\i dv di dx3y' dxdy
V =
9 W j
ex
Moi quan he ket hop thiet lap dira tren luat Hooke b6i phuong trinh sau:
a = D„(z)(E„-ZK) T = D,(z)v (i2a,b)
04.2016 anSllf IIBX1113
=[v vT
D.(z) =
0 0 ( 1 - v ) / 2
(15) 2 { H - v ) [ 0 I j
Trong dd k la he sd hieu chfnh cat.
4 . M o h i n h phSn tfch dao d o n g td do cua t a m FGM 4.1.HdmdgngMK
Phuang phap MK ddpc ddng de xay ddng ham dang va cac dao ham [24-26]. Gia t h i l t ham phan b d u ( x , ) dupe xap xi trong m i i n con i i , sao cho n , c f i . G i i sd rSng cac gia tri cua ham sd dddc ndi suy dda tren cac g i i trj tai ehe d i l m ndt x , [ i e [ l , n ] 1 v d i n i i tdng sd diem nut trong m i i n n , . Ham ndi suy MK u" ( x ) , Vx e f i , ddpc xac djnh nhd sau:
hay
u''(x) = [p^(x)A + r"(x}B]u(x)
u''{x) = £ a ) , [ x ) u ,
Trong ddCD|{x)lci h i m dang MK, ddOc x i c dinh nhdsc
®,(x) = [p^(x)A + r'(x)B]
Ma tran A, B ddpc dinh nghia nhd sau:
A = fP^R"'Pr'p^R"'
(16)
(18)
(19) (20) B = R"'(I-PA)
Trong do I la ma t r i n dan vi, vec to p[x) ik da thde vdi m h&m cP s6:
p'{x) = [pi(x),pj{x),p,(x) p„(x)] (21) Cu the, ddi vdi ma t r i n P kich thudc n x m , cac gia tri cua h i m co sd
da thde (13) dupe eho bdi nhusau-
•pi(x,) pj(x,) -• p „ ( x , ) "
PitXj) P3(Xj) - • p^(Xj)
P = (22)
_ p , { x j p,(x,) - p j x j _ Vec t o r(x) trong phuang trinh (16) ddOc dinh nghia nhU sau:
r^{x) = [R(x,.x),R(x„x),....R(x„.x)] (23) Trong d d R ( x , , X j ) l a h i m tdong quan gida cac cap cOa
nut \ va X, nd ddpc bieu hien bang cac phdong sai cua c i c trUdng gia t r i u ( x ) : R(X|,x,) = eov[u(x,),u[X|)]va R(x„x) = eov[u(X|),u(x)]. Cd nhieu each de xac djnh h i m R(x,,X|) nhung phdong phap h i m Gauss la phdPng p h i p thddng sd dung vi tinh dPn g i i n , hieu q u i
R ( x . , x , ) . e - * <«) Vdi: i;| = | ] x | - x , [ | , v i 9 > 0 la he sd tUong quan. Trong bai bao nay sd
dung p ^ ( x ) i a m d t h i m b a c h a i n h d s a u :
p'(x) = [ l , x , y , x ' , y ' , x y ] (25) Ngoai ra, ma trinR[R(x,,X|)]^ ddpc bieu didn dUdi dang tudng
minh nhusau:
R[R(X„X,)]
1 R{x„Xj) •• R(x,,xJ R(Xj,x,) 1 •• R{Xj,xJ
Ddi vdi b i i toan tam FGM, khdng chi dao h i m b^c 1 ddpc sil dyng' m i cdn dao ham b i c 2 cda h i m dang cung duoc thiet lap nhusau:
U^) = 'Zp>M)f<, + Zr,^(x)B^ (27, i t
<!>,,„(=<) = SPl,i,('<)A„ + | : r , , [ x ) B „ (28) C S n l d u i / i n h h u d n g c d a h f sd tuang quan 6 ddi vdi ham dang lird'
rang. Mdt trong nhdng diem quan trpng nhat eda h i m dang MK, 36 l i sd hdu t i n h chat Kronecker's delta. Dieu n i y se loai bd nhdng t r d ngai d i n g k l nhat cua hau h i t c i c phuong phap khdng ludi khi i p d i t di^u kien bien de giai bai toan cd hoc. D l chihig minh cho dieu n i y , chiing [a k h i o sat lai ham dang MK x i c dinh bdi b i l u thde [18)
<^,(x,) = Z P ) ( ' ' j ' A , i + Z ' ' k ^ ' S t j (29) Hay bieu thde (29) cd t h i v i l t dUcri dang sau:
[ 0 , ( x , ) ] = PA+RB (30) Trong d d ma tran A,B,R v i P d d p c djnh nghTa bdi cdng thilc
(19)(20) v i (22). Thay eong thde (20) vao (30) ta ddpc:
[ 0 | ( x , ] ] = PA + R R - ' ( l - P A ) - l (31) B i l u thde (31) d i n d i n tinh ch^t Kronecker's delta x i c dinh bdi bilu
thde (32).
, , . f l k h i i = j (32) ' ' [ O k h n ^ ^ j
N g o i i ra, ham n d i s u y M K s d hdu tinh nhat quan, nghTa l i e d t h i xiy ddng lai b i t ed h i m cd bac t h i p han. De don g i i n , thudc tinh n i y cdth^
t d m t i t nhu sau' N l u u, dat dupc tif da thilc cd b$c nhd han hoac bSng m nghTa la
u = Pa (33) trong dd, P dUoc xae d[nh t i l edng thde (22) va a la hi sd bat ky, thi
sUxap xl dd l i chinh xac SdxSp xT cua trUdng c h u y i n vj nhUsau:
u N x ) - p " [ x ) a = u(x) (3^) D i e biet, neu sd d u n g ham p(x) ! i ham t u y l n t i n h khi x i y ddng
ham dang MK t h i tat c i h i n g sd, sd hang t u y e n t i n h cd the x i c dinh lai h o i n t o i n :
| ; W x ) = l , X < t , ( x ) x , = x , X < h ( x ) y , = y (35) Mat khic, mdt trong cac y l u t d quan trong ddi vdi phdOng phip khdng Iddi la mien anh hddng, trong do b i n kinh m i i n i n h hddmg ddi^c ddng de x i c dinh sd Idpng cac nOt rdi rac trong pham vi mien nfli suy dang x^t. Ban kinh mien i n h hddng d ^ d d p c x i e dinh nhdsau:
d, = a d . (36)
Trong d d a la he sd cila mien gia d d , thdng thddng a nSm trong khoang t d 2.0 den 3.0. Gia trj d, l i chieu dai dae trung cho k h o i n g cich c i c ndt vdi diem dang x4t.
4.2. Cdc phuong trinh rdS rac
Nhdng chuyen vi trong he tpa dp t d n g q u i t trong mat phSng glila duoc xap xf theo bleu thde (17), trong d d
o-.[u> «• „ : w;]' M
" , - h ' i w „ w „ f (38) Thay bieu thde (17) vao bieu thde (1 l.a,b,c) nhan dupe
e o ^ S s r u , K = 2 B r u . V = t B ; u , p g .
D" = J D^(z)dz ^ " " I 2D„(z)dz (43a,b)
D'' - J z^D,„(z)dz (44)
T r o n g d d :
Br=
B;"
•fri.. 0 0 0 0 <(.„ 0 0
*l.y K 0 0 To 0 0 /f,^
~ [ o 0 V (ti,^, B[' =
0 0 (|)|„ 0 0 0 ^lyy 0 0 0 »h^ 0
V d i b i i t o i n p h a n t i c h tinh, dang y l u duPc bieu dien n h d sau:
j'5e"'DEdfi-f-j5Y^D'Ydfi = J 5 ( W i ^ + w J d f i j^.,) Trong do f la lUc phan bd deu tren don vi di6n tich, va
TD'"
=w 0-r B D" -• I D3(z)dz {42a,b,e)
5. Ket quS s6
Trong phan n i y , ilng x i l dao ddng t d do eda tam FGM vdi chi sd n suy giSm thay ddi cdng vdi eie dieu kien bien k h i c nhau ddpc k h i o sit dda tren md hlnh phan tieh k i t hpp gida ly t h u y l t 5-FSD vdi phuong p h i p khdng ludi MKG (S-FSD-MKG). Luoc do bac 2 Gauss 4 x 4 dupe sd dung trong phdong p h i p khdng Iddi MKG de tich phan dang yeu. Oieu kien bien cua tam dupe k^ hieu n h d sau: gdi tua don gian (S), ngam (Q, va t d do (F). Cae dieu kien bien n i y dUoc i p dat thdng qua e i c phUcmg trinh nhusau [27]
(i) Canh bien gdi tUa ddn:
Thay t h i b i l u thde (39) v i (42a,b,c) v i o bieu thde Error! Reference source n o t found.) b i i t o i n phan tich tTnh ciia t^m FGM cd the viet lai nhusau:
Kd = F (45a,b) Trong dd ma tran do cdng trong he toa dp tdng t h i x i c dinh nhusau:
V^e t o lUc dupe x i e djnh nhu sau:
F = J f N d n y^ M^^[o 0 ^^ ^ j T j ^ ^ j
Bii toan 2: Khao sat i n h hUdng eda dieu kien bien, cht sd do suy g i i m n den dd vdng tai diem gida cda tam FGM /H'AljQ hinh vudng cd t f sd a/h = 100. Thude tinh vat lieu edaAl l i : v ^ = 0 . 3 , E ^ = 7 0 G P a eila AljOj l i : V;=0.3,E^ =380GPa.Tam sd dung sd lddng d i l m nut l i : 1 3 x 1 3 . He s d h i ^ u ehinh c i t k = 0.8601. Phdong p h i p khdng Iddi MKG s d d u n g e i c thdng s6: a = 3,9 = 3 . K h i o s i t dp vdng eua tam FGM neu tren dddi tac dung cua t i i trpng phan bd: P = l N / m m ' . Dp vdng tai
_6W|,
_dv/,
" ax O , t a i x = 0,a
- 0 , t a i y = 0 , b .
u = v = w. =—
(ii) Canh bien gdi tUa n g i m :
^b _ _ ^s _ ^%
dy ^ dx dy taix = 0,a v a y = 0,b.
Bii t o i n 1:Giasdtam FGM hlnh vudng, san xuat t d v i t lieu fi|7sQ^-^, cd t f sd gida e h i l u dai v i chieu d i y ^ h , chl sd dd suy g i i m n thay ddi.
Thude tinh vat lieu cda Al l i : v „ = 0.3,E^ = 70GPa cua ZrO, - 1 la:
Vj = 0.3,E, =200GPa.Tam s d d u n g sd Idpng diem ndt l i 1 3 x 1 3 . H e s d hi^u ehinh c i t k = 0.8601. Phddng p h i p khdng lUdi MKG s d d u n g cac thdng sd: a = 3,9 = 3 . K h i o s i t dd vdng cda tam FGM neu tren dudi tac dung cda tai trong phan bd: P = 1 N / m m ' . O d vdng tai diem gida ddoc
100w^E„h^
chuSn hda bdi:v
- , 2 ( , - - . Ldi g i i i ve dd vdng dat ddpc t d eae phuong p h i p khae nhau dupc t h i hien d b i n g 1. Ket q u i cho thay cd sy phu hpp, sU chinh xac gida phuong phap d l nghj (S-FSD-MKG) vdi c i c phuang p h i p dua tren ky t h u i t sd khae ddi vdi eae dieu kien bi^n, t y le gida e h i l u d i i v i chieu d i y k h i c nhau cua tam FGM. Mat khic, khi d i l u ki$n bi^n thay ddi t d (SSSS) sang (SFSF) thi d p cdng eda ket cau se giam, dan den sd gia tang ve d d vdng eda tam. N g o i i ra, khi tang chi sd d d suy g i i m n , dp vdng cOa tSm se gia tang.
diem gida ddPc chuan hda bdi: v _ ] 0 w ^ E ^
Pa" 0 sanh vdi eie ket q u i dat dupe t d cac phuong p h i p k h i c nhau ddpc t h i hien d b i n g 2. Dda v i o k i t qua eda b i n g 2, ta t h i y khi dieu kien bien thay ddi t d CCCC sang SCSC, SSSS v i SFSF, bi^n dd cda dp vdng diem gida tam FGM t i n g dan d i u dan. Khi chl sd n gia tang thi bien d d cda dp vdng diem giQa tam FGM eung tang dan. K i t qua ve g i i tri dp vdng dat 3uae t d phuang p h i p de xuat (S-FSD-MKG) rat phd hpp vdi eae ldi giai dat dupe tir phuang phap sd khae.
6. Ket luan:
Hinh 2. flo wng ciia tam mong A^AIjC^, chl SD suy giam n=2cacaieu kien bien khae nhau (a) SFSF, (b) SSSS, (c) SCSC, (d) CCCC
04.2016 smniRnsx 1115
B i n g 1. Dd vdng chuan hda cda tam fi^ZK^-l vdi sil thay ddi ve tJ' sd ^ h , cht sd d d suy g i i m D i l u kien bien
SSSS
SFSF
^ h 5
20
100
5
20
100
Phuang p h i p S-FSDT based IGA[27]
FSDT based IGA[27]
S-FSD-MKG S-FSDT based IGA[27]
FSDT based IGA[27]
5-FSD-MKG S-FSDT based IGA[27]
FSDT based IGA[27]
S-FSD-MKG S-FSDT based IGA[27]
FSDT based IGA[27]
S-FSD-MKG S-FSDT based IGA FSDT based IGA[27]
S-FSD-MKG S-FSDT based IGA[27]
FSDT based IGA S-FSD-MKG
n = 0 0.1717 0.1717 0.1777 0.1440 0.1440 0.1 S07 0.1423 0.1423 01490 0.5083 0.5089 0.4939 0 4 6 1 4 0.4615 0 4483 0.4584 0.4584 0.4454
n=0.5 0.2324 0.2324 0.2402 0.1972 0.1972 0.2058 0.1949 0.1949 0.2036 0.6918 0.6926 0.6717 0.6319 0.6321 0.6135 0.6281 0.6281 0.6098
n = l 0.2719 0.2719 0.2803 0.2310 0.2310 0.2403 0.2284 0.2284 0.2378 0.8099 0.8108 0.7858 0.7404 0.7406 0.7183 0.7360 0.7360 0.7139
n = 2 ' 4 0.3115 -i.
0.3115 : 0.3204 :-ia 0.2628 0.2628 0.2728 02597 0.2597 0.2698 0.9247 0.9258 0.8968 0.8420 0.8422 08164 0.8367 0.8367 0.8112
B i n g 0.2 Chuyin D i l u ki^n bi4n SFSF
SSSS
S C S C
C C C C
VI chinh giCia t^m FGM c Phuang phap S-FSDT[27]
FSDT[27]
S-FSD-MKG S-FSDT[27]
FSDT[27]
S-FSD-MKG S-FSDT[27]
FSDTI27]
S-FSD-MKG 5-FSDT[27]
FSDT[27]
S-FSD-MKG
D a/h = 100 n = 0 1.4302 1.4302 1.3896 0.4438 0.4438 0.4648 0.2096 0.2097 0.2066 0.1384 0.1384 0.1370
vdi c i c dieu kien bien k h i c nhau n=0.5
2.2062 2.2062 2.1405 0.6846 0.6847 0.7132 0.3232 0.3234 0.3158 0.2135 0.2135 0.2104
n = l 2.8692 2.8693 2.7781 0.8904 0.8904 0.9204 0.4204 0.4205 0.4053 0.2776 0.2776 0 2719
n=2 3.6770 3.6770 3.5525 1.1411 1.1411 1.1696 0.5387 0.5389 0.5123 0.3557 0.3558 0.3460
n=5 4.3483 4.3483 4.2042 1.3494 1.3494 1.3873 0.6372 0.6375 0.6088 0.4208 0.4209 0.4103
n=10 4.7740 4.7740 4.6257 1.4816 1.4816 1.5357 0.6996 0 7000 0.6777 0.4621 0.4622 0.4535
Bii b i o d i de xuat mdt md hlnh phan tich tTnh cua tam FGM sd d y n g md hinh p h i n tich k i t hpp gida ly thuyet S-FSD vdi phuang phap khdng lUdi MKG (S-FSD-MKG). Cac vi du sd ve phan ti'ch dao ddng t u do cua tam FGM ddpc thde hien nham so sanh Idi g i i i dat ddpc vdi Idi g i i i t d phuong phap so khic. Cac yeu t d i n h hddng den dao ddng t U d o cua tam FGM c h i n g han nhU: dieu kien bien, chi sd do suy giam n cung dUoc k h i o s i t . K i t qua cho thay viec sd dung md hinh d l xuat mdi vdi so an sd it han, nhUng van cho k i t qua chinh xhe ddi vdi t i m mdng v i tam day. Mat dd nghien cdu d day chl thde hien vdi nhdng b i i toan t u y l n tinh, tam cd hinh dang vudng, tuy vay phdong p h i p nay eung cd the thyc hien ddi vdi nhdng bai toan phi tuyen va ddi vdi tam ed hinh dang bat ky.
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