25:
Tuyen tap cdng trinh Hdi nghi Cff hoc todn (pidc Ky niem 30 mini Vien Cff hge vd 30 mint Tiip chi Cff hge Hd Noi, ngiiy 8-9/4/2009
Phan tich co' che pha huy dam be tong cot thep theo co' hoc ran niVt va CO' hoc pha huy be tong
Tran The Truyen, Ho Xuan Nam
Khoa Cdng trinh, Tnedng Dgi hge Giao thdng Van tdi.
Email; lranthetruven(a).uct. edit, vn. hoxiiannam2005S2(d) yahoo, com
Tom tat. Bdi bdo trinh bdy cdc tinh todn phdn tich phd hiiy ddm be tong cot thep dan gidn theo hai tiep can phd hiiy khde nhau:(i). Tiep cdn theo ca hoc phd hiiy ddn be tdng vdi gid thiet moi trudng tinh todn lien tuc, (ii). Tiep can theo ca hoc rgn niet be tdng vdi gid thiet moi trudng tinh todn khdng lien tuc. Cdc ket qud cd dicgc theo hai tiep can tinh todn khde nhau ndy se dugc phdn tich vd ddnh gid khi so sdnh vdi cdc ket cpid thuc nghiem.
1. Dat van de
ling xir CO hge eiia be tdng kha phirc tap vi be tdng la vat lieu rat khdng ddng nhat dugc cau thanh bdi mdt hdn hgp gdm eae vat lieu cd dac tinh co ly khac nhau. Viec ap dung ly thuyet dan hdi de tinh toan eae ket eau be tdng ehi hgp li khi be tdng chiu tai nhd trong giai doan dan hdi. De khiic pbiic ban che nay, cac md hinh phi tuyen phan tich sir lam viec ciia be tdng dudi tac dung eiia tai trgng va cae tac ddng ciia mdi trudng da duge nhieu tae gia de nghj tren co sd cae gia thiet ve irng xir phi tuyen ciia be tdng ca trirde va sau khi dat den irng suat gidi ban va dac biet la xet den cac dac diem khdng ddng nhat va phi tuyen vat li d nhung vimg be tdng bj mem hoa do xuat hien cae dudng nirt nhd. Trong cac tiep can phi tuyen phan tich sir pha hoai ciia be tdng, tiep can theo ly thuyet co hge pha hiiy (damage mechanics) va tiep can theo ly thuyet CO hge ran niit (fracture mechanics) la nhu'ng phuong phap tien tien va iugc sir dung rdng rai hien nay tren the gidi.
C> tilp can thir nhat [9], cac md binh dien hinh cd the ke den gdm: Mazars (1984), Simo &
Ju (1987a, 1987b), Pijaudier-Cabot & Bazant (1988-1989), Jirasek (1996, 2004)... Cac md hinh nay da duge dl nghj dira tren gia thiit trudng tinh toan la lien tiie ddng thdi ed xet den anh hudng eiia su xuat hien ciia cac dudng nirt nhd trong cae vimg chju lue bat Igi ciia be tdng lam trilt giam do cirng khiln be tdng bj mem boa. Nbuge diem ciia cac md hinh theo tiep ciin nay lii khdng xet den bien dang du trong be tdng.
Ci tilp can thir hai [9], cac md hinh dien hinh cd the ke den gdm: Dugdale & Barenblatt (1960-1962), Hiller-Borg (1984), Bazant (1983).... Cac md hinh nay ed xet den viing pha hoai cue bd dau dudng nirt FPZ (Fracture Process Zone) eho phep md ta chinh xac ve dieu kien lan truyen eiia eae dudng nirt trong be tdng va sir tap trung bien dang trong eae viing bj pha buy.
Vdi eae gia thiit ve mdi trudng tinh toan cd the la khdng lien tuc (nhu md hinh dudng nirt ao PCM Fictions Crack Model Hiller Borg (1984)) hoac lien tuc ylu (nhu md hinh dai nirt CBM- Crack Band Model Bazant (1983)), viec lap trinh tinh toiin sd bang phuong phap lien tuc nhu phuong phap phSn tir huu ban tuong ddi khd khan vi ket qua phu thude rat nhieu vao ludi phSn tir [6]. Vi thi mgt sd md hinh phi tuyen xap xi (Approximate models) nhu TPM
254 Trdn Thi Truvin, Hd Xudn Nam
(Two-Parameter Model - Jenq& Shah (1985)), SEM (Size Effect Model - Bazant (1984, 1987)) hay ECM (Effective Crack Model Karihaloo (1992, 1995)) da duge dl nghj va ap dung cho cae tinh toan don gian hda [1]. Trong thuc te, cae md hinh nirt xap xi nay dugc dimg ket hgp vdi cac ly thuyet co' ban eiia ket eau be tdng edt thep de phuc vu cac tinh toan ung dung.
Trong nghien ciru nay, tuong ung vdi mdi tiep can nhu tren, chiing tdi tien hanh lira chon mgt md hinh dien binh de tinh toan phan tich sir pha hoai eiia mdt dam be tdng edt thep dan gian. Vdi tiep can theo ly thuyet co' hge pha hiiy, chiing tdi chgn md hinh Mazars khdng cue bo [7], [8]. Vol tiep can theo ly tbuj'et co hge ran nut, chirng tdi chgn md hinh phi tuyen xap xi TPM [1]. Mo hinh dau da duge cac tac gia dua vao ma ngudn phan mem phan tich phan tii hiru ban Lagamine [10], md hinh sau dugc eae tac gia lap trinh tren Matlab 6.0 [11]. Cac ket qua eo duge se duge danh gia va so sanh vdi cac quan siit thuc nghiem
2. Tom lu'oc cac mo hinh sii' dung de tinh toan
2.1. Md hinh phd huy don khdng cue bg Mazars
Trudng irng suat duge bieu dien qua bien dang ed xet den tham sd pha hu}' D:
a i j = ( l - D ) C , . „ 8 „ (1) Cudng do bien dang cue bd duge dien giai bang bien dang tuong duong;
Vdi: <£,> = 0 neu £i < 0
<ei> = £,• neu e, > 0;
e, la eae bien dang chinh theo eae phuong i (i =1, 2, 3).
Sir phat trien ciia pha buy duge dac trung bdi bam nguong pha buy:
f ( 8 , D ) = 8 - K ( D )
K(D=0) = EDO ggi la gidi ban pha buy ban dSu.
Bien pha buy toan phan bSng tdng hgp biin pha buy keo (D,) vii khi nen (Dc).
D = at Dt + a e Dc
EK
(2)
(3)
(4)
Hinh 1. Mo hinh Mazars bieu dien trong khong gian img suat
Phdn tich ca chi phd hiiy ddm be tdng cot thep theo ca hoc rgn niet vd CO hoc phd hiiy bi tdng
Qua trinh gia tang eiia bien pha buy phu thuge va gia trj ciia biin dang tuo'ng dirong:
Neu e > EDO :
D = a,.D, + a^.Dc = a,.D, + (l-a,).D, D... = 1 - -
a.-(Z<^' r^'^--
exp[B,^(8-eDo)]
255
(5) (6)
(7)
- Neu e < EDO :
D = 0. (8) trong do: D, va D^ la eae thanh phan pha buy khi keo va khi nen; SDQ la nguong pha buy ban dau;
At,c, Btc la eiie tham sd pha buy khi keo va khi nen; P la tham sd pha buy e4t.
Ham phat trien pha buy dugc viet dudi dang dieu kien Khun-Tucker cd dang nhu sau:
f < 0 ; D ' = 0 ; f D ' = 0 (9) Tiep can khdng cue bd duge dua vao de xet den hien tugng tap trung bien dang trong cae vimg
bj phii hiiy eiia be tdng. Khi dd, bien dang tuong duong cue bd s^ dugc thay biing ddi irng la bien dang tuo'ng duong khdng cue bd 8 theo cdng thirc dugc ehuiin hoa nhu sau:
e(x) = f T ( x -s)s(s)ds
VM)L
V,(x)= J ^ ( x - s ) d s
Vdi phan bd viing pha hiiy cd the chgn nhu dang phiin bd chuan Gauss sau day [8]:
^ ( x - s ) = exp ' 4 ( x - s ) ^ ^
(10) (11)
(12) Ic la chieu dai dac trung cho biet kich thude eiia vimg pha hiiy FPZ.
2.2. Mo hinh niii xdp xi TPM
Md hinh nirt xdp xi TPM dugc dl nghi bdi Jenq & Shah (1985) tren co sd ly thuyet co hge ran nirt phi tuyin nham kbSc phuc cac khd khan khi lap trinh tinh toan theo eae phuang phap sd.
phat trien duAing nut tn/dt dinh
JJZ
CIOD 1
~> 7;::>^
II
1
IJ
\ Sy" VC3 v^)
^ . X
Hinh 2 - Dudng nirt cd hieu va quan he P-CMOD theo md hinh TPM (CMOD - Do md rdng mieng dudng nirt)
256 Trdn Thi Truyin, Hd Xudn Nam
Chiiu dai dudng nirt thuc tl a„ trong ket eiiu be tdng iugc thay the bang dudng nirt cd hieu
«(, trong kit cau ed irng xir dan hdi tuong duong. Qua trinh lan truyin cac dudng nirt bat dau khi tai trgng dat den gia trj tdi ban P^ax (ehilu dai dudng nirt ed hieu dugc tinh tuong irng vdi P,,,;,^):
P = P,„a.. (13) Khi dd: K = Kc va CTOD = CTODc. (14)
Vdi: Kc la he sd cudng do irng suat tdi ban dugc xac djnh bang thuc nghiem theo co' bgc ran nirt tuyen tinh (LEFM) khi thay a = ae; CTODc la do md rdng tdi ban ciia dau dudng nirt dugc tinh vdi a = a^.
Trong eae tinh toan irng dung, md hinh TPM thudng dugc sir dung ket hgp vdi cac ly thuylt CO' ban ve sir lam viec ciia cac ket eau be tdng edt thep [1]. Vd'i bai toan udn dam be tong edt thep, su lan truyin ciia dudng nirt xien xay ra do hieu irng tdng cdng lan truyen nirt theo Mode 1 va Mode II gay ra bdi ddng thdi cae irng suat phap (Vdi he sd cudng do irng suat K,) va irng suit cat (Vdi he sd cudng do ung suat K,,) trong dam. Dudng nirt tdi ban gay pha hiiy dam xuat hien khi xay ddng thdi hai dieu kien:
- Cd su lan truyin nirt mat dn dinh theo dieu kien nirt tdng hop ciia Mode I va Mode II :
K=K, + K „ = K c (15) - Be tdng d vimg chju nen bi ep vd. Dieu kien bi ep vd ed the theo tieu chi pha hoai deo
Mohr-Columb hoac Drueker-Prager cai tien [1], [6]
Tren Hinh 3, dam bi pha hiiy theo mgt dirdng nirt tdi ban ed tga do tdi han Xcr tuo'ng irng vdi tai trgng pha hiiy dam P|„„,, khi dudng quan he "Tai trgng — be tdng bj ep vd" (a) gap dudng quan he "Tai trgng - lan truyen nirt mat dn djnh" (b) [1].
p ,
>
1
(a)
C b ) ^
y
X/
1 1
Hinh 3. Quan he giiia tai trgng P va tga do dudng dudng nui x 3. Vi du tinh toan
Dam be tdng edt thep don gian khdng ed cot dai sir dung be tdng ed cudng do chju nen lang tru f e = 25 MPa (M25) cd cdu tao nhu tren Hinh 4.
Ciic kich thude co ban ciia d§m eho d Bang 1 sau day.
Bang 1. Cac kieh thude co ban eiia dam be tdng cot thep L(mm)
1500
S(mni) 1400
B(mm) 100
W(mm) 150
c(mm) 25
Phdn tich ca chi phd hiiy ddm be tdng edt thep theo ca hoc rgn niet vd ca hoc phd hiiy bi tdng
257
1 ' -
1 -
A ,1
"
rv'
1D12 Hinh 4. Dam be tdng edt thep gian don khdng ed cot dai
Cae dac trung co hge eua be tdng gdm [12]: Md dun dan boi E = 26422 MPa; He sd Poisson v = 0.2; Nguong pha-buy ddn eoo = 0.00006; He sd pha buy khi keo: A, = 0.6; B, = 15000; He so pha buy khi nen: A^ = 1.5; Be = 2000; He sd pha buy khi cit: [3=1; Chiiu dai dac trung ciia viing pha buy ^ = 0.02 m. He sd cudng do irng suat tdi ban Kc = 1.4.
Cae cot thep D12 cd dac trung co bgc [11], [12]: Md dun dan hdi E = 2,1.10' MPa; Gidi ban ben fy = 370 MPa; Bien dang tdi ban 8c = 0.27.
3.1. Tinh todn phd huy ddm theo mo hinh don khong cue bg
Dam md phdng dugc chia ludi vdi 800 PTHH 8 mit (Hinh 5), ludi phan tir mjn din tir dSu diim vao giira. U'ng xir ddn khdng cue bg dugc giin cho be tdng, irng xir dan hdi deo tuyet ddi duge gan eho thep, lien ket giii'a be tdng va edt thep dugc gia thiet la tuyet ddi. Qua trinh md phdng eung dugc tien himh trong dieu kien khdng che do vong dam ciia dam. Ket qua md phdng diroc bieu dien tren Hinh 6 va Hinh 7.
I j i — ^ n ^ ~ I I I ! !! !! li i
Hinh 5. Ludi phiin tir huu ban md phdng dam (800 PTHH)
Ket qua tinh toan dam be tdng edt thep theo tiep can pha hiiy ddn eho thay: Vj tri va phuo'ng cac viing pha hiiy va tap trung bien dang (Hinh 6) (tuang irng vdi vj tri va phuong ciia dirdng nirt tudng tugng) tren dam md phdng thu duge gan gidng vdi vj tri va phuong ciia cac dudng nirt thuc quan sat duge tren dam thuc nghiem. Tren Hinh 7, ket qua md phdng nam cao hon so vdi ket qua thuc nghiem, dieu nay dugc giai thich bdi gia thiet lien ket tuyet ddi giua be tdng va edt thep cung nhu cac gia thiet don gian hda khac khi tinh toan nen dain md phong ben hon dam thuc nghiem. Khi tai trgng P xap xi 5000 N, dudng cong quan he "P-V" bat dau ddi hudng, day chinh la thai diem xuat hien vimg pha hiiy dau tien trong be tdng (tuong irng vdi khi bien dang tuong duo'ng dat den nguong pha hiiy dau tien), dieu nay dugc lam rd khi chiing tdi ghi nhan tren dam thuc nghiem la thdi diem phai thao tenso'inet do bien dang vi ed sir xuat hien cac dudng nirt nhd. Day cung lii thdi diem danh diiu su bat dau ciia irng xir phi tuyen ket eiiu gay ra bdi phi tuyin vat lieu. Tai trgng gidi ban bat dau pha hiiy dam hoan toan P,„ax (tuong irng vdi thdi diem dudng nirt tdi ban pha hiiy dam (Hinh 9) xuat hien va lan truyen nhanh) ciia dam md phdng kha sat vdi kit qua thirc nghiem.
258 Trdn The Truvin, Hd Xudn Nam
ft
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Hinh 6. Phiin bd eae vimg pha hiiy (a), vimg tap trung bien dang (b) va cac dudng nirt thuc tren dam thi nghiem (e)
0,0000 0 0010 0,0020 0 0030 V(m)
Mo phong C— Thi nghiem
Hinh 7. Quan he Tai trgng - do vdng "P-V" ciia dam 3.2. Tinh todn phd huy ddm theo md hinh TPM
Cac birdc tjnh toan lan truyen nirt theo md hinh TPM ed kit hgp vdi eae nguyen ly co ban eiia ly thuylt uon dSm be tdng cot thep kha phire tap va dai ddng [11]. Sau day chirng tdi chi trinh bay cac dac diem tinh toan chinh va kit qua thu dugc. So do tinh toan dudng nirt tdi han giiy pha hiiy dSin dudi tai trgng P duge bilu diin nhu tren Hinh 8.
Tren Hinh 8, Fs la lire cang ciia edt thep khi be tdng bj pha hoai theo mdt dudng nirt cd toa do la X va gdc nghieng la 9. (Gdc nghieng 9 la mdt ham ciia tga do x). Cac budc tinh toan pha
Phdn tich co chi phd hiiy ddm bi tdng cot thep theo ca hoc rgn niet vd ca hoc phd hiiy bi tdng
259
hiiy dam duge thuc hien bang each lira chgn eae cap gia trj khac nhau eiia x va G vdi cac sd gia Ax va A6 trong khoang tir gdi ke dam din giua nhjp (vj tri dat tai trgng P). Gia trj lire P„„„ pha hiiy dam thu Augc la gia tri ?„,^^ nhd nhit thu dugc tir eae gia trj P„„„ tuong irng vdi cac cap (x, 6). De cd the chgn dugc nhung cap gia trj (x, 9) nim gan vj tri dirdng nirt tdi ban (tuong irng vdi tai trgng pha buy dam P,„ax nhd nhat) vdi mue dich la giam khdi lugng tinh toiin, eae quan siit vl dac diem phan bd vj fri va phuong cac dudng nirt tren cac dim thirc nghiem da duge irng dung.
Fs(x+iAx) Hinh 8. So dd tinh toiin dudng nirt pha hiiy dam theo md hinh TPM Cae ket qua thu dugc duge bieu dien trong Bang 2.
Bang 2. Ket qua tinh toan dam be tdng edt thep Ket quii
Tai trong tdi han P,,,,,^ ( kN) Goc pha hoai 0 (°)
Chieu dai dirdng nut tdi han a^r (cm) Tga dp duong nut tdi han x^ (cm)
Mo hinh TPM 19.026 45.1668
12.4 58
Thi nghiem 21.50 45.16 11.58 57
-riz?
^ $it^''f^-'7C^m>^'^^^^^>^>^^'^w^'s?K^'h ^^yt-j.' ''' *
Hinh 9. Vj tri va phuong ciia dudng nirt tdi ban pha hiiy dam
Kit qua tinh toan pha hiiy dam theo tiep can eo' bgc ran nii't ket hgp vdi ly thuyet udn dam be tdng edt thep (Bang 2) cho thiiy vj tri va phuong eiia dudng nirt tdi ban pha hiiy dam duge xiie djnh tuong ddi chinh xac khi so sanh vdi thuc nghiem. Tuy nhien, ket qua ed duge khdng cho bilt thdng tin vl phan bd eiia cac dudng mit nhd trudc khi diim bj phii hiiy theo dudng nirt tdi ban nhu d tinh toan md phdng theo tiep ciin thir nhiit. Ket qua phii hiiy dam tren Hinh 9 dugc
260 Trdn Thi Truyin, Hd Xudn Nam
md ta kha id vdi trudng hop dang xet (nghTa la dam be tdng edt thep khdng cd edt dai), nen pha hiiy cit xay ra rd rang theo mdt dudng nirt tdi ban.
4. Ket luan
Kit qua tinh toan phi tuyin dim theo hai tiep can khac nhau da eung cap nhiing thdng tin khac nhau \ I CO ehi pha hiiy eiia mgt dam be tdng cot thep don gian. Vdi tiep can theo co hge pha hiiy ddn, kit qua thu duge kha sat vdi thuc nghiem, tuy nhien chi md ta dugc giai doan trudc khi xuit hien dudng nirt Idn gay pha hiiy dim. Trong khi do, tiep can theo co hge ran nirt eho phep danh gia co chi xuit hien va lan truyen dudng nirt tdi han pha hiiy dam, vj tri va phuong ciia dudng nirt tdi ban duge xac djnh ro va gan vdi eae gia trj tbiie nghiem, dii vay tiep can nay khdng cho bilt phan bd cac dudng nirt nhd trude kh! pha hoai dam. Nhu vay, mdi tiep can co uu nhiigc diem rieng, can can nhae khi su dung trong thuc te.
Tai lieu tham khao
[1]. B. Karihaloo. Fracture mechanics & structural concrete, Longman Scientific & Technical, New York Wiley, 1995.
[2]. Bazant. Z & M. Jirasek. Nonlocal integral formulation of plasticity and damage: Survey of progress, JE.M,ASCE,2002.
[3]. D.R.J.OWEN, A.J. FAWKES. Engineering Fracture Mechanics: Numerical Methods and Application, Pineridge Press Ltd, Swansea, UK, 1983.
[4]. Faustino.S.J, Wison.S.V Damage modelling of reinforced concrete beam. Advance in Engineering software, accepeted August 13"', 2006.
[5]. M. Jirasek. Nonlocal damage mechanics with application in concrete, Revue Europeenne de Genie CivH, 8, 2004.
[6]. M. Jirasek. Plasticity, damage and fracture, Fragments of lecture note, UPC, Barcelona, 11, 2002 [7]. Mazars.J. Application de la mecanique de rendommagement au comportement nonlineaire et a la
rupture du beton de structure, These doctoral d'etat, Universite Paris VI, 1984.
[8]. Pijaudier-Cabot, Bazant.Z. Nonlocal damage theory. Journal of Engineering Mechanics, vol 113, 1987.
[9]. Tran The Truyen, Nguyen Viet Trung, Cac ni6 hinh ung xii' co hoc cua vat lieu be tong va lira chon mo hinh t6i uu diing trong tinh toan irng dung. Tap chi 1<HGTVT, 4, 2007.
[10]. Tran The Truyen, Nguyen Viet Trung, Robert Charlier, Tap trung biin dang va mo hinh khong cue bo trong ni6 phong irng xu be tong theo li thuygt pha huy don, Tgp chi Cau dudng. 8, 2007.
[11]. The Truyen Tran, Etudes du inodele de rupture par propagation des fissures dans les poutres en beton arme, Master Thesis, Universite de Liege, 2004.
[12]. Tran The Truyen, Nguyen Viet Trung, Ly thuyet va thuc nghiem xac djnh cac tham s6 cua mo hinh pha huy don be tong, irng dung vao phan tfch co chi pha buy ciia cac bo phan kit ciu cong trinh giao thong, Tgp chi KHGTVT, 10, 2008.
