Mo phong sir phat trien vet nuTt trong ket cau dap be tong bang phUorng phap phan tilr huTu han md rong
Simulation of crad propagation in concrete dam structures using extended finite element method
Ngay nhan bai: 13/02/2017 Ngay sira bai: 8/3/2017 Ngay chap nhan dang: 5/4/2017
T 6 M TAT:
Viec danh gia miic do an toan ciia cac k^t c3u dap be tong doi hoi sU phan tich chinh xac cac vet nilt. Sif lan truyen vet niJt n^u khong diipc Idem scat tot cd th^ dan den sil mat on dinh tnfot, sU chia cat nen dap hoac lam gia tang dong tham tif h6 chiia.
Tren cO scf ilng dung n^y, mot m6 hmh tinh todn so sii dung phildng phap phan lii hflu h ^ md rong (eXtended Finite Element Method, X-FEM) ket hop vdi phildng phap Level Set (Level Set Method, LSM) ddQc x§y diJng nham nghien ciJu sd lan truyen vet nilt trong dap be tong. Mot vi du dien hinh sit dung mo hinh tinh nay ddac trinh bay trong bai bao de t h i hiin quy trinh tinh va kha nang i2ng dung cua phildng phap. Cac ket qua tinh toan thu diloc cho thay mo hinh tinh c6 the dil dodn chfnh xac quy dao vet mlt Nghien cilu se gop phan vao cac no liic hien tai trong viec tinh toan cd hoc niit gay 6 cic k^t cau dap be tong.
TU kh6a: dap be t6ng, sfl phat trien vet mit, cd hpc mlt gay, phifdng phap phSn tii hflu h^n md rOng, phUdng phap Level Set
ABSTRACT:
Assessing the safety of concrete dams requires an accurate prediction of cracking.
Uncontrolled crack propagation could lead to the sliding instability, the separation of dam base or the increase in seepage from the reservoir. In this paper, an algorithm where the eXtended Finite Element Method (X-FEM) is coupled with the Level Set Method (LSM) is proposed to study crack propagation in concrete dams. A validation example is presented to demonstrate the analysis procedure and capacities of the method. The results obtamed show that the proposed method enables an accurate prediction ofthe crack trajectory. This research aims to contribute to the continuing efforts on mastering the mechanics of cracking in concrete dam structures.
Keywords: concrete dams, crack propagation, fracture mechanics, eXtended Finite Element Method (X-FEM), Level Set Method (LSM)
TS V6 Thi Tuyet Giang
Khoa Ky thuat Xay dilng - Trtlcfng Dai hoe Bach Khoa, DHQG-HCM, Viet Nam TS Nguyen Vo Trong^
Khoa Ky thu^t Xay dUng - TriTdng Dai hpc Bach Khoa, DHQG-HCM. Viet Nam
Vo Thj Tuyet Giang, Nguyen Vo Trpng
1.Tdng quan
Su nghi§n ciiu ve nut trong cae dap bi tong thudng tap trung vao hai van di ehinh: thuat toan phan tich va mo hinh mo phdng s6 hap 1#.
Bi d y doan tien trinh phat trien vet niJt trong cac dap be tong nay, ngUb\ ta sC( dung hai phuong phap tong qu^t trong phan tieh ph^n tU hClu han la vet nilt rdi rae (discrete crack) va vet niJt phan tan [smeared crack). Ddi vdi bai toan cO hoc ran niit khi sCr dung phuong phap phSn ti!f hOli han, vet ndt phu thuoc vao ludi (mesh) cua mien tinh toan va khi vet nut lan truyen, cSc thuat toSn hieu chinh ludi (remeshing algorithms) can difOc thi^t lap. Cong viec nay gSy ra sU bat loi khi sCfdung phuong phap phan t d hdu han. Vdi phuong phap ph^n t d bien (Boundary Element Method, BEM), bien cua mien nghi§n edu dUOc ehia thanh cac phan tCr. Phuong phap phan t d bien cd Uu diem la viec phan ehia phan t d d o n gliin, thdi gian tinh ngSn; tuy nhien so vdi phuang p h i p phan tdhflu han, phuong phap phan t d bien can cac ky thuSt dac biet 3i gidi quyet eSc mien nghien edu c6 tfac diem phi tuyen, khong dong nh^t hoac kh6ng d i n g hudng.
Nghien cUu trong b^i bao nay phat trien mot mo hinh toi Uu de mo phdng sU lan truyen v^t ntJt bSng phuang phap phSn t d hdu han md rdng (PTHHMR) ket hop vdi phuong phap Level Set l/u di^m chinh cCia phuong phap PTHHMR la v^t niit duoe the hien ma khdng can p h i i chia ludi no [1].
Phuong p h i p PTHHMR cd the md phdng cScti^n trinh lan t r u y i n vet nUt vdi duy nhSt mot lUdi tinh toSn va CO the cho de ket quS chinh xic v^ cac he sd cudng do dng suSt (Stress Intensity Factors, SlFs). Phuong phap Level Set H phuang phSpsfi sd dung lUdi ed dmh dUOc p h i t t r i l n bdi Osher va Sethian [2] de md phdng sii phM tnen cOa cac mat bien va hinh khdi. Phuong phap nay rat thich hop cho cac bai toan cd bien d?ng Idn ho|c cSi bai toan cd bien di ddng nhU bai toan ve lan truyen vet ndt. Be md ta mdt vet ndt, hai ham Level Set duoc sd dung [3]. Hal hhm nay du^c cap nhat trong moi bUde tfnh toan nhSm dua ra hinh dang vet ndt mdi.
Trong bSl bao nay, mdt md hinh tinh toan so sd dung phUOng phap PTHHMR va phUOng phap
Level Set duoc xay dung nham nghien edu sd lan truyen vet ndt trong dSp b e t d n g Su khdng Ii6n tyc ciJa chuyen vj do vit ndt gay ra dUOc dUa vSo bdi ham Heaviside suy rpng v a s u b d sung eScviJng lan can dinh v^t ndt, nhSm tang tinh chinh xae eda co hoc ran ndt dan hdi.
Phan tiep theo cua bai bSo duoc trinh bay nhu sau. 6 phan tiep theo, tong quan ve phuong phap PTHHMR va phuong phap Level Set cung nhu ly t h u y l t tinh toan sU lan truyen vit ndt duoc gidi thieu. Cac k i t q u i md phdng sd ve sUlan t r u y i n v i t ndt do mdi trong mdt mo hinh dsjp thu nhd da cd sSn v i t ndt duoc trinh bay d Phan 3 vdi cSc t h i o luan. Va eudi cCing, d PhSn 4, mot vai ket luan cting hudng phat tri^n nghi§n edu t i l p theo duoc dde k i t iai.
2. Phu'cmg phap s6
2.1. T^ng quan ve phirong phap PTHHMR va phu'cmg phap Level Set
Trong phuong phap Level Set, bien dl ddng duoethe hien la tap mde khdng (zero level set) cua mdt ham cd bae eao h o n . O l md t5 mdt vet ndt, hai ham Level Set dUOc sd dung. HSm Level Set phap tuyen (normal level set, Isn) t h i hien khoang each d i n mat ndt vi h i m Level Set tiep tuyen (tangent level set, 1st) the hien khoang each d i n dinh v i t ndt.
Phuong phap PTHHMR cho phep md phdng sy phat t r i l n vet ndt ma khdng can dieu chinh ludi tinh toan ban d^u. Su khong lien tuc cua chuyen vi do vet ndt gay ra dUoc dUa vao bdi ham Heaviside suy rdng va sUbd sung cac VLing lan can d l n h v l t n u t , nhSm tSng tinh ehinh xac eda CO hoc ran ndt dan hdi Cdng thdc tinh xip xi chuyen vi u''{x) tai d i l m x duoe the hien bang Cdng thdc (1). Thanh phan dau tien trong Cdng thdc (1) la thanh phan chuyen vi co b i n (li§n tuc). ThSnh phan t h d hai dUa v I o sU khdng lien tuc cua chuyen vi khi xet cac niit bi chia dt bcri vet ndt (Heaviside enrichment) va thanh phan t h d b a dUa vao dac diem phi t u y l n khi xet cae niit lan can vet ndt (asymptotic enrichment hay crack-tip enrichment). Hinh 1 a the hien mot vet ndt vdi dc bSc t y do bd tro eho cic niit duoc md rdng. LUu ^ rSng vet ndt di ludn qua cac ndt va rd r i n g vet ndt khdng duoc chia lUdi [4].
u^(x}= 2 1 a,<P,[x)+ X bja)j(x>H[lsn(x)) +
Z Z<^*°*><''1''"<'^"<'<'''^(''"
(1)Vdi; - a, la cae bae tU do eiia chuyin vi lien tuc gan vdi niit i, - 01 la h i m dang t u y l n tfnh cua niit i,
- bj la cac ble tU do bd tra gan vdi ntit j , - c ° l a c a e b a c t u d o b d t r o g a n v d i n i i t k .
H i m H{x) la ham Heaviside, khong lien tue qua be mat vet ndt. K h i o sat mdt niit bj ehia cat bdi vet ndt, H(l5n(x)) = 1 ddi vdi eac nut nam d phia l5n(x) > 0, trong khi dd H{lsn(x)) = -1 ddi vdi eac nut n i m d phJa cdn lai, cd Isn(x) < 0.
H i m F" , a = 1.4, dUoc viet nhu sau [5]:
V p s i n | , V p c o s ^ , T P sin—sin (p, . / p cos—sin ip
CIc toa dd cue dja phuong (p,<|i) e d t h e duac viet dUdi dang cle level set tai dinh vet ndt nhusau (xem Hinh l b ) ;
p = \/lsn'+lst^;<p = t a n ' ' -—
tftt
I _ /
ttftt
— *^
(a) (b) Hinh 1. Vet nOt vdi car bktu do bo tro cho cac mit dirocmS rgng (ky hieu trtin dai dpdio car nut diToc mb rong b^i ham Heaviside va ky hieu vuong dai dien dio cac mit duoc mdrong hoJ dc ham phi luyen) (a), he loa do Caitesian tong the (r,z), he toa do Cartesian dia phuong (n,t), he toa do cue dia phuong 1 p , iji) va cat level set (b).
Hai dang md rdng eda nut tren dinh vet ndt duac t h i h i l n d Hinh 2.
Odi vdi dang md rdng t d pd (topological ennehment), chl cd mdt niit tren dinh vet ndt dUcJc md rdng (Hinh 2a). Ddi vdi dang md rdng hinh hoc, tat d cac nut xung quanh dinh vet ndt trong pham vi dUdng trdn cd ban kinh rmax duoc md rdng (Hinh 2b). LUu y rang sd luong cac ndt dUcfc md rgng se tang khi ludi tinh toan dUOc chia mjn [4].
/- /-' T
^-i-f-
-'^ ^ /
- T _
_ 1
-4- -
(a) (b)
Hinh 2. Hal dang mil rong ciia mit tren dinh vet nut: dang mdrong to p6 (a) va dang mti rong hinh hoc (b).
2.2. l y thuyet tinh toan sU lan truyen vet nut - Thuyet dng suat phap theo phifong tiep tuyen cUc dai
Erdogan va Sih [6] dua ra t h u y l t dng su^t phap theo phUong t i l p t u y l n cUe dai de tinh t o l n sy lan truyen vet niit. Theo thuyet n l y , vet mlt se p h i t t n I n khi dng suat phap theo phuong tiep t u y l n oee cUc dai. Gan dinh vet ndt, trudng dng su^t dan hdi duoc cho bdi Cdng thdc (4).
- ^ / 2 ^ ^ " \ 2 cos - e KLcosM - e - - K I L s i n e (4) Gdc phat trien cua vet ndt duoc tinh t d viec lay dao ham Cdng thdc (4) va dUcJc the hien d Cong thde (5).
- 2 t a n "
4 Kll -sign(Kll)
,Kii; (5)
Vdi Kl, KH I I cac he sd cudng do dng suat.
SU tinh t o l n suat giai phdng nang lucmg dUoc thUe hien b i n g each sd dung phUOng phap G-theta [7]. PhUOng phap n l y dua tren cac tich phan mien v l ph^p lay dao ham Lagrange eda the nang, tUong dng vdi mdt trudng v l n tdc md rdng v i t ndt l o 6. De tfnh toan elc he sd cUdng do dng suat, phUOng p h i p G-theta dUoc md rong vdi d^ng song tuyen tfnh cua suit g i l i phdng nang luong [5]. Trong trUdng hop bai toan 2D, chi duy nhat mdt trudng 9 I I can t h i l t
3.Md hinh tinh toan
De xay dUng md hinh tinh toan vet ndt, ehung tdi da p h i t trien mdt chuong trinh tinh dua tren phuong phap PTHHMR trong Code.Aster [8], mdt phan mem ma nguon md dUoc p h i t trien bdi Tap d o l n Dien lUc cua
04.2017BU!inffi[fSSai|15:
Phap (EDF R&D) t d nam 1989, Mdt bai t o l n md phdng 2D duoc trinh b I y trong muc n l y nham chdng minh k h i nang dng dung ciia t h u i t t o l n dUa ra va minh hoa dd chfnh xae eda su k i t hop phuong phap PTHHMR va phuong phap Level Set trong viec md phdng sU lan truyen vet nut trong dap be tdng
Bai toan n l y md phdng sU lan truyen vet ndt do mdi trong mdt md hinh dap thu nhd (tU mdt dap be tdng eao 96 m) da cd s i n vet ndt. Md hinh thu nhd nay da cd duoe elc k i t qua thf nghiem thu dUOc cua Carpintien et al. [9] va elc k i t q u i md phdng sd bang elc phuong p h i p khae (nhu bang phuang phap phan t d hdu han [9] va phuang p h i p phan t d bl^n [10]) n l n thuan tien cho viee so sinh c i c ket q u i tinh t o l n .
Ap lUe nudc duoc tao ra t d co cau truyen dong vdi k h i nang t l i la 2000 kN va duac t i c dung len mat thuong lUu dap [9] Ap lUc n l y duac phan phoi t h i n h bdn luc t i p trung cd dd Idn tang theo chieu s l u nUdc.
Hinh ve 3 minh hoa kich thUde d i p , cac I p lUc nudc tUOng dUong v l vi trf ban d^u cua vet ndt. Cung vdi muc dieh so sanh cac k i t q u i eCia md hinh tinh vdi elc ket q u i thi nghi§m v l k i t q u i md phdng sd b^ng cac phuang phap khlc, v i t ndt ban d l u x u l t hien d m l t thuong luu v l d vi tri cich chan d i p mdt khoang each bang 1/4chieu cao dap.Chieu dai v i t ndt ban dau la I S e m . Vet ndt n l y khong phu thuoc vao ludi tinh t o l n dua tren viec sddung cac h i m level set p h i p tuyen v l tiep tuyen.
Hinh 3. Mo hinh dap, cac ap lUc mfiic tuong dirrnig va VI tri ban dau ciia vet mit CIc tham sd v i t liSu I I md-dun d i n hdi b i n g 35700 MPa v l he sd Poisson bang 0.1 LUdl sddung trong tinh toan bao gom 62128 phan t d tam gilc. Kich thUde Idn nhat ciia mdt mat ludi la 1 cm, LUdi duoc chia min d vung gSn dinh vet ndt (khoang elch duac d i n h gia bdi g i l tri eCia cac level set) tai mdi bude tinh lan truyen vet ndt de dam b i o viec tinh toan chfnh x l e cac he sd cUdng do dng suit. N l n luu y r i n g v i t ndt khong duac chia ludi trong t o l n bd q u i trinh tinh b i n g phuong phap PTHHMR.
Do tang trudng v i t ndt Idn nhat duac thiet lap la 4 cm
Trong mdi budc tfnh t o l n sU p h i t tnen v i t ndt, mot quy trinh tinh l i p duac thuc h i l n n h I m d i m b i o dat duoc su hdi ty h o l n t o l n , Su hdi tu duae djnh nghia la phep g i l i trong dd k i t q u i thu duac ciJa h§ sd cudng do dng suit hoi tu v l cung gia tn trong moi bUde tinh t o l n n l y va cle k i t qua c6 su dao ddng dn dinh trong phep thd cle lUdi tinh toan dUoc chia min k h l c nhau.
Su lan truyen vet ndt bao gdm budc cap nhat cle level set va bUdc dinh nghTa dinh v i t ndt mdi. Bleu do t i l n trinh chi tiet cho mdt md hinh tfnh sy lan truyen vet ndt bang phuang phap PTHHMR vdi Code_Aster duac the hien d Hinh 4, Tien trinh n l y bao gdm sau budc ehinh' (1) tao ludi tinh t o l n (mesh), (2) dmh nghia v i t ndt (crack definition) b i n g cac ham level set, (3) dUa vao thuat toan tinh t o l n lan truyen v i t ndt bang phuang phap PTHHMR (finite element enrichment), (4) chay md hinh tinh {computation resolution), (5) tinh t o l n eae k i t q u i he sd eUdng dd dng suat, quy d^o va chieu dai vet ndt (post-processing SIF, propagation) va (6) b i l n thj cac ket q u i tfnh.
K i t q u i md phdng gdm 23 bUde tinh toan lan truyen v i t ndt. Tdng thdi gian chay chuong trinh tinh ia 21s. Dieu nay cho t h i y vdi chuong trinh tinh tren, thdi gian tinh toan se giam d i n g ke khi so sanh vdi thdi gian tinh t o l n bang each g i l i elc phuong trinh Hamilton- Jaeobi, bang elc phuang phap CO dien hoae bang c i c phuang phap sd khac.
Hinh 5 the hien k i t q u i quy dao vet ndt sau 23 bude tinh toan lan t r u y i n v i t ndt. Cac k i t qua nay tUOng ddng vdi cac k i t q u i thuc nghi&m cua Carpintieri et al. [9]
(vung duoc gach cheo 'Test 2" t h i hien dinh v i t ndt di xuyen qua c h i l u rdng d i p trong thi nghiem; cung nhU tucjng ddng vdi cac k i t q u i t d md hinh tinh b i n g
phuong phap phan t d hiiii hgn (dUdng "Ref. [Sf, Carpintieri et al [9]) v l bSng phuang p h i p phan t d b i l n (dUdng "BEM", Chahrour v l Ohtsu [10)) (Hinh 6), Nhu v l y , cd t h i dua ra ket luan vi m l t t u o n g ddng ket q u i glOa sU k i t hop phuong p h i p PTHHIVIR va phUong ph^p Level Set vdi cic phuong p h i p khlc trong viec md phdng su lan truyen v i t ndt trong dap bitdng
Tren Hinh 7, cae h i so cudng do dng suat Kl va Kll duoc bieu diSn theo budc tinh t o l n lan truyen vet ndt. Cd the n h i n xet r i n g cic gia tn Kl d dinh vet nut tang khi chieu d l i v i t ndt t i n g v l elc gia tn Kll nho hfln nhieu so vdi eae gia tn K! Sy lan truyen v i t ndt I I khdng t u y l n tinh v l dang md rdng (opening - mode I) la dang lan t r u y i n vet nut chil y l u . LUu
•^ rSng Kll cd the cd gia tri nhd (duong hole am) khi vet ndt lan truyen, phu thudc vao kich thudc tang trUdng eCia v i t ndt da chon. D i l u nly c6 the g i l i thich t d viee tfnh toan hudng vet ndt cua bUdc tinh k l t i l p dUflc x i c dinh t d cae g i l tri Kl v l Kll cto bUde tinh trUde do.
Visualisaiion Hinh 4. Bieu do ti^n trinh chi tiet cho mot mi hinh linh su lan truyen vel mil bing phuong phip PTHHMRvfliCode flster.
Hinh 5 . Quy dao lan tmyen vet ni)t tfnh bang sU ket hop phuong phap PTHHMR va phuong phap Level Set
54»OniKlffB]i 04.2017
Hinh 6 . Quy dao lan tmyen vet ndt thu Suae bang thuc nghiem (Test 2), bSng phi/dng p h i p phan t£r hull han (Ref 18]) va bang phuong p h i p phar tilbiSn (BEM) [9,10],
^vuu,
\n
Hinh7.Cac he so ci/cmg do ling suat Klva Kll bieudier theo budc tmh toan lan truyen vet niJt.
4 . Ket luan
Trong bai bao nay, mdt m d hinh tfnh toan so sd dung phuang p h i p PTHHMR va phuang phap Level Set duoc xay dyng nham nghien edu sU lan truyen vet ndt trong dap b l tdng. Lfu d l l m ehfnh cCia phuong p h i p PTHHMR la vet ndt dUOe the hiSn mdt c i c h dde lap vdi ludi tfnh toan, do dd khdng can phai chia lai lUdi khi vet ndt lan truyen. SU khdng lien tuc cCia chuyen vj do vet ndt gay ra dUOc dua vao bdi ham Heaviside suy rdng v l sU bd sung cac viing l l n can dinh vet mlt, n h I m t i n g tinh ehinh xle ciia CO hoe ran ndt dan hdi. Di g i l m chi phi tinh toan va tang dd chfnh xac cua cle budc tfnh lan truyen vet nut, vj tri dinh vet ndt va hinh d^ng v i t ndt duoe dinh nghia b i n g hai ham level set ma cae thdng so ciia chung chi phu thudc v i o eac d i e trUng hinh h^c eda b l i t o l n ,
CIc k i t q u i tfnh toan thu duac cho thay md hinh tfnh edthe dudoan chinh x i c quy d^o v i t ndt, chdng minh k h i nang dng dyng cua t h u i t t o l n dUa ra va minh hoa do chinh xae eda sU ket hop phUemg phap PTHHMR va phUOng phap Level Set trong vifc md phdng su lan t r u y i n vet ndt trong d i p b l tdng. Vdi md hinh tinh nly, thdi gian tfnh toan se g i l m d i n g ke khi so s i n h vdi thdi gian tinh t o l n bSng cich g i l i dc phuang trinh Hamilton-Jacobi, bSng cae phuong p h i p co dien ho|e bang cae phuong p h i p so khac. Oieu nay cho thay t r i l n vong cua md hinh t o l n dUa ra ddi vdi viec nghien edu sU lan truyen vet ndt.
Nghien edu t r l n gdp phan v i o eac no lyc h i f n tai trong viec tinh t o l n ea hoc ndt gay d elc ket cau dap be tdng. Trong cle nghien edu tiep theo, sy tinh toan va md phdng sU Ian truyen vet ndt se duac p h i t trien ddi vdi mdt sd dap be tdng thUe te.
5. LiA dm Un
N g h i e n e d u n l y S u o e t l i t r o b d i T r U d n g O a i h o c B i c h K h o a - D a i h o c Q u d c G i a t h a n h p h d H o C h f M m h t r o n g k h u d n k h d d e t l i c a p t r u d n g v d i m a s d d l t a i la ' T - K T X D - 2 0 1 6 - 7 6 " . C I c t a e g i l c u n g t r a n t r o n g e l m o n s u h o p t i c v d i T r u n g t a m m d h i n h h o a v a m d p h d n g ( M o d e l l i n g a n d S i m u l a t i o n C e n t r e , M A S C ) , D a i h o e M a n c h e s t e r , A n h t r o n g n g h i e n e d u v l
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