v_ J
iNGHE t t t iO A N H GIA B O T I N C A Y
CAU DAN THEP BANG MO PHONG MONTE CARLO
BACH VAN SV PHAM X U A N TUNG T R A N QUANG HUY Khoa Xay dung, Oai hoc Nha Trang
Tom tat: Phan tich kit ciu dwa tren nin tang xac suit cho cong trinh ciu da dwac nghien cwu trong mot thai gian dai tren thi glal. Cu thi nhit cua dang phan tich xac suit/rui ro la thiet ki theo he s6 tii trong va s&c khang (LRFD) nhw trong tieu chuin thiit ki ciu cua My (AASHTO) hay Viet Nam (TCVN 11823.2017). Bai bao nay giai thieu ting quan phwang phap phan tich do tin cay ket ciu bing md phong Monte Carlo, day la phwang phap a mire do cao han so vol phuang phap LRFD vi nd chinh xac ban va cd t h i dua ra dup'c xac s u i t pha huy cua k i t c l u d y a tren cac yiu t6 ngiu nhien cua Iwc va kha nang chiu Iwc cua kit ciu. Ngoai ra, bai bao cung cap mot vi du cu thi hwang din each thua phan tich do tin cay cho mot kit ciu ciu dan thep nhip gian dan di minh hoa cho phwang phap.
TIP khoa: Do tin cay kit ciu, mo phong Monte Carlo, ciu dan thep, xac suit pha huy
Abstract: Probabilistic-based design for bridge structures has been developed worldwide for a long time. The design guidelines of American (AASHTO) or Vietnamese (TCVN 11823:2017) using the Load and Resistance Factor Design (LRFD) concept are essentially based on the risk-based design format. In this paper, reliability analysis for structures using Monte Carlo simulation (MCS) is introduced. MCS is a higher form of reliability assessment than LRFD because it is more accuracy and can predict the probability of exceeding the failure of a structure based on the uncertainty of load and resistance random variables. In addition, this paper provides an example of reliability analysis for a single steel truss bridge to illustrate the applications of MCS.
Keywords: Reliability assessment, Monte Carlo simulation, steel truss bridge, probability of failure
1. DAT VAN DE
Danh gia an toan d u a tren n i n tang xac s u i t (hay ggi chung la danh gia do tin cay) d u a e s u dung rong rai d Chau A u , Uc va My Ngoai viec danh gia v i mat an toan, thi danh gia do tin eay cung r i t hiru ich khi ap dung cho viee danh gia han e h i tai trgng, phat sinh chi phi gia eudng c i u , van hanh hay ngirng van hanh c l u [1]
Cdng trinh c l u ludn phai ddi mat vdi nhieu tac dgng n g i u nhien va r i t khd cd t h i d u doan, v i du n h u s y thay doi ciia tai trong xe, tac ddng ciia moi t r u d n g . va dap cua tau be vao m i tru c i u . hay iiin ciia nen dat lam cdng trinh bi chuyen dich va m i t su'c chiu tai. Tren t h i
gidi hien nay da nghien eii'U nhieu khia canh cho cdng trinh cau, n h u viec xac djnh he sd tai trong va siFC khang cho epc khoan nhdi [2], danh gia riii ro cho he t h i n g cgc- d l t bang phuang phap mat phan irng (response surface method) [3], hay danh gia dp tin eay eho eau d i m I [4].
Bai bao nay gidi thieu phuang phap danh gia do tin cay b i n g md phdng Monte Carlo va hudng d i n each thirc thue hien viec danh gia thdng qua mpt vi du eu the ap dung cho c i u dan thep nhjp n g i n Bai bao dup'c b6 cue nhu sau' Muc 2 mo ta khai mem v i danh gia dp tin cay cho k i t c i u cdng trinh va gidi thieu cac phuang phap xae dmh dp tin cay Mue 3 gidi thieu v i md
phdng Monte Carlo, Muc 4 md ta tLPng b u d c trinh t y phan tich xac suat xay ra c h u y i n vi vuat qua c h u y i n v| cho phep cua mgt c l u dan thep nhjp dai 20m Phan tich ket hop md phdng Monte Carlo va phuang phap phan tie hiru ban (PTHH) b i n g lap trinh Matlab K i t luan va hudng phat trien nghien ciKU trinh bay trong Mue 5.
2. DANH GIA DO TIN CAY CHO K i l C A U
2.1. Nguyen ly v i phan tich do tin cay
Gd tin eay eua k i t e l u dai dien thdng qua hai thdng s6 ca ban la tai trgng (S) va site khang (R). Tuy nhien, ca hai thong sd nay deu khd d u doan vi chiing la nhirng b i i n ngau nhien va tinh khdng c h l e chan (uncertainty) ciia chung duae bieu d i i n thdng qua ham phan p h i i xac s u i t (probability density function, ky hieu la pdf), f |,-) va /"^ Is) K i t c l u an toan khi R > 8.
Nguoc lai, pha hoai cua k i t c i u dug'c djnh nghTa la khi R < S Xac s u i t pha hoai, p dupe xac dmh nhu sau [5]
P,=P(R<S) = P{s{R,S)<Q] = \\fjT,s)drds ( 1 )
Trong dd fif.ii/'V la ham phan bd xac suat hai chieu cua hai b i i n n g i u nhien va Q la m i i n pha hoai.
Xac suat pha hoai trong trudng hap nay chinh la dien tich t i i p xue gitra hai dudng cong phan phdi xac suat, the hien la vung gach spe trong Hinh 1
Trong trudng hap xay dyng ham trang thai gidi ban ehung, cac thdng s i ve tai trgng va sifc khang ed the bieu d i i n d u d i dang cac b i i n n g i u nhien, ky hieu la x , tic
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• # # # # KHOA HOC - CONG NGHE
Sirt ktiang (R)
Hinh 1. Nguyen ty dinh Iwang rui ro
do ham trang thai gidi han se cd quan he tuo'ng ii-ng vdi mdi tieu ehi v i trang thai (hay mgt tinh nang mong dai) nhu sau.
x ( x ) = , v ( x . x , , . , X J (2) Trong dd X la mdt bg cac bien ngau nhien ciia k i t e l u , n la t i n g s i b i i n ngau nhien, va ham g(X) diing d i xac djnh tinh nang hay trang thai eua ket cau, g(X) ~ 0 dup'c xem la trang thai gidi han.
Hinh 2. Khai mem ve trang thai giai han
Hinh 2 trinh bay khai niem v i trang thai gidi ban, trong dd d u d n g phan chia vung an toan (g(X) > 0) va VLing khdng an toan {g(X) < 0) d u a c djnh nghTa la mat pha hoai (failure surface) hay la mat trang thai gidi han g(X) - 0, Phuang trinh (1) luc nay ed the trinh bay d u d i dang bien n g l u nhien X^
P,=\... \ / , ( . T , , / , , . . , x J r f r ( 3 )
Trong dd, fy,[^\'^2'-'^t,) la ham phan phdi xac s u i t da chieu (joint pdf) cua eac b i i n n g i u nhien
2.2 cac phu'cng phap danh gia dp tin cay
Nguyen ly ehinh ciia phan tich dp tin cay la giai dupe tieh phan da chieu cua phuang trinh (3). Tren thye te, viec tinh toan xac suat pha hiiy la rat khd va ddi khi khong the thye hien d u a c cho cong trinh thuc t i do khdng xac dinh dupe qui luat phan p h i i , Cae phuang phap phan tfch thu'dng chia thanh hai loai d y a tren ham trang thai gidi ban hien (explicit) hoae an (implicit) [6].
Cae phuang phap ciia ham hien gdm-
- Ldl giai chinh xac cho b i i n cd phan phoi t y nhien hoae Idgarit t y nhien
- Phuang phap MVFOSM (Mean Value First Order Second Moment) - Phuang phap FORM (First-Order Reliability Method)
- Phuang phap SORM (Second- Order Reliability Method) - Phuang phap md phdng Monte Carlo (Monte Carlo Simulation) Cac phuang phap eiia ham I n gdm:
- Phuang phap md phdng Monte Carlo (Monte Carlo Simulation) - Phuang phap md phdng Latin Hypercube (Latin Hypercube Simulation)
Phuang phap mat phan ung (Response Surface Approach) Mo phdng Monte Carlo dupe lya chgn cho nghien eu'U nay vi kha nang ap dung rong ciia phuang
phap. Ngoai ra, phuang phap nay duae xem la cho ra k i t qua chinh xac nhat khi phan tich vdi mot so luo'ng m i u ldn cd the dai dien cho s i ddng, Chi tiet phuang phap mo phdng Monte Cario trinh bay trong Muc 3,
3.M0PH6NGMDNTECARL0 Md phdng Monte Cario t h u a n g dup'c phat t n i n de kiem chirng cho cac phuang phap danh gia dp tin eay khac vi nd la phuang phap dan gian dai dien cho t y nhien va cd tinh chinh xac cao n i u phat sinh du so luang b i i n n g l u nhien, Ngoai ra, ky s u hoae nha nghien ciru chi can trang bj lupng k i i n thicc c a ban lien quan d i n xac s u i t va t h i n g ke d i dmh luang dp tin cay hay rui ro cho cac he t h i n g ky thuat phifc tap [7].
Nguyen tae ciia md phdng Monte Carlo la phat sinh N m l u dpc lap cua vec ta b i i n n g i u nhien (X) d y a tren ham phan phdi xac s u i t da chieu biet trudc f (.v)- Tip mdi b i i n n g l u nhien, x^, ta xac djnh d u a c k i t qua eiia ham trang thai gidi ban ^ . f x ) - N h u d a d i cap ben tren, n i u g ( x ) < 0 ta gpi la pha hoai. Vay c u tneo djnh nghTa co ban ciia xac s u i t , ta cd xac s u i t pha hoai chinh la ty s i giua tong s i pha hoai ( N , ) chia cho t i n g so m l u {N) eua phep thii' nhu cdng thiKc (4) Mipo do chinh xac cua md phdng d u a c danh gia theo edng thue (5) Dp ehinh xae ciia k i t qua phu thugc vao s i lugng vdng lap md phong, d u a e danh gia dya tren khoang tin cay 95% ciia xac suat pha hoai d y k i i n .
N, Phan tram sai so.
(4)
(5)
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-. NCHE § MM S a d i md phdng Monte Carlo trinhbay trong Hinh 3. Viee phat sinh bien ngau nhien ciia md phong c i n d y a tren mpt phan p h i i eu the B i t d i u b i n g viec phat sinh bien n g i u nhien don vi, a,, tip 0 den 1 theo phan phdi d i n g dang. Cac cong cu may tinh hien nay d i u lam dug'c, vi du nhu MS Excel diing ham rand ( ..) Sau dd, eac bien n g l u nhien dan vj viFa phat sinh dup-e ve d u d i dang phan p h i i tieh luy (cumulative distribution function hay viet t i t la cdf), Ap dung p h u a n g phap phan p h i i tich luy ngup'c (inverse cdf), nghTa la xem cdf cua b i i n n g i u nhien c i n tim bang vdi cdf ciia b i i n n g l u nhien dan vj vira phat sinh, '^x(^',) ^ ' ^ i , ta tinh d u p e b i i n n g l u nhien c i n tim [7], X = f - ' (fl, )
(6) Trong trudng hp'p b i i n n g i u nhien can tim cd phan phdi t y nhien, thi gia tri e i n tim xac dmh theo cong thiFC sau:
X^ - / / ^ -HO-^ -O ' ( f l j (7)
Trong do /^x va "^x l i n lug't la gia trj trung binh va dp lech c h u i n da biet t r u d c ciia b i i n c i n phat sinh, O la cdf eua phan p h i i chuan dan gian (^ = 0 va a = 1), va fl^ la b i i n ngau nhien dan vj.
4. VI DU PHAH TICH CHO MOT CAUDAHGIAHD0HHH!PL = 2OM 4 . 1 . Mo ta k i t cau
Trong nghien cicu nay, mdt nhip e l u dan thep dan gian chiu tai trgng tTnh (gia dinh nhu tai trpng ddng) dyyac l i y lam vi du. Muc dich eua vi du nay la giup ngudi dpc hieu ro each t i i p can tinh toan dp tin cay b i n g mo phong Monte Carlo cho mot cdng trinh cu the. He dan ehu ehiu lyc true t i i p \\s tai trpng ddng la loai thep djnh hinh chCp C, cac thanh dan con lai sir dung loai thep binh I nhu mo ta trong Hinh 4.
Tieu c h u i n thep ASTM A709, kich t h u d c t i i t dien thanh dup'c tong hpp trong Bang 1
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Md hinh md phong Monte Bicn ngau TIIHCH, X Carlo
Hinh 4. Mo hinh ket cau nhip cau dan thep L - 20m Bang 1. Tiit dien ciu kien cua kit ciu theo tieu chuin ASTM Loai thanh ASTM
inxib/ft (mmxkg/m) C10x20(C250x29,8) W10x22 (W250«32,7) W10x19(W250x28.4)
Dien tich (mm^j AI A2 A3
3794 4190 3630
d (mm)
254 258 260
b (mm)
70 146 102
(mm) 11,1 9,1 10,0
(
(mm) 9,6 6,1 6,3
(cm*) 3284,1 4926,6 4012,7
C-shape beam
4.2. Phipo-ng t r i n h trang thai gio-i han
Phuang trinh trang thai gidi han xay dung dua tren chuyen vi eho phep (allowable deflection) gitpa nhjp do tai trgng ddng gay ra theo d i x u i t ciia AASHTTO [8] hay TCVN 11823:2017(9], U = L/800 = 2000/800 = 2,5 cm. Phuong trinh cu t h i nhu sau:
Hay trong trudng hap cu the cua VI du nay la
g{u) = 2,5-u, (cm) (9)
g(w]-L/800-w, (8)
Gia trj chuyen vi giiua nhip u^ dup'c giai thdng qua md hinh dan hai ehiiu (2D) bang phuang phap p h i n tic hiru han, vdi s y thay doi ciia eac b i i n n g l u nhien dau vao gom ngoai lyc, md dun dar\(h6i, va t i i t dien k i t c i u . Cac b i M n o a u nhien dup'c t h i n g ke trongjMf • \^
eaui-
^ # # # # KHOA HOC - CONG NGHE
^ i L .
2, trong do he sd bien tham khao Shah P M . va edng s y [1]
S a n g 2. Thong s6 diu vao ngiu nhien Bi§n
ngau nhien Ngoai ll^C 1 Ngoai lire 2 Mo dun dan hoi Tiet dien thanh AI Tiet dien thanh A2
Gia tn
210
310
2,1x10"
3794
4190 Dan
VI
kN
kPa
mm^
He so bien
0,2
0,06
0,04 Loai phan phoi
TIP nhien
Tv nhien
Til' nhien Tiet dien I 3630 i , i ,
'thanh A S ' ' I I I
B i n g phuang phap md phdng Monte Carlo md ta ben tren, cac b i i n n g l u nhien trong Bang 2 dupe phat sinh thanh 10.000 000 bo k i t qua thong qua gia trj trung binh, he s i bien va dang phan phdi eua nd Sau dd, cae thdng sd nay d u a c gan vao md hinh p h i n tir hu'u han de giai bai toan tim c h u y i n vi giu'a nhip u, Theo nhu phuang trinh (8) hoae (9), neu g(u) < Ota dmh nghTa k i t c l u bl pha hoai. Trong thuc te, ed t h i ket cau van ehua pha hoai, tuy nhien viee dmh nghTa nay se dam bao cho ket cau dug'c an toan dua tren yeu c l u v i c h u y i n vi cho phep ciia k i t c l u Can hieu rd la viec pha hoai hay khdng pha hoai la phu thugc vao dinh nghTa eiia ngudi phan tich va co t h i chua phai la pha hoai vat ly ciia k i t e l u thye t i
4.3. K i t qua p h a n t i c h xac suat pha huy
K i t qua phan tfch d i m d u a c cd 227 666 k i t qua n g i u nhien bi pha hoai tren tong s i 10 theu k i t qua phan tich Dua theo p h u a n g trinh (4), ta tinh d\yac xac s u i t pha hiiy
la 0,02277 (hay khoang 2,3% tren tong so m l u thic). P h i n tram sai s6 tinh dupe t u cong thu'C (5) la khoang 0,41%. Hay cd the phat b i i u la ed 95% kha nang xac suat pha hiiy xay ra trong khoang 0,02277 + 9,34x10"^ ung vdi 10 trieu md phdng.
Ddi vdi cdng trinh thuc t i eo tinh chat phifc tap, ta can dmh nghTa nhieu phuang trinh trang thai gidi han khae nhau de danh gia toan dien do tin cay ciia k i t c l u img vdi mdt yeu c l u an toan eu t h i . Viee lya chpn dup'c gia tri bien n g i u nhien phii hap can dya vao k i t qua do dae thye nghiem hoae md phdng dang tin cay de danh gia cdng trinh thye t i ,
5. KET LUAN
Bai bao nay md ta phuang phap md phdng Monte Cario phuc vu danh gia do tin cay eho k i t e l u cdng trinh thye t i Ngoai ra, cd phan tieh mdt vi dy cu t h i eau dan thep nhip gian dan de huo'ng dan each thu'C thuc hien phuang phap Trudng hap phan tich cu the xac suat ket eau c i u dan L = 20 m b;
chuyen vi v u a t qua gidi han cho phep (hay edn gpi la xac s u i t pha huy nhu djnh nghTa cua nghien eii'U nay) la khoang 2,3%, ii*ng vdi thong s i dau vao ngau nhien nhu md ta trong nghien cii'U nay Md phdng Monte Carlo la phuong phap eho ra k i t qua ehinh xac nhat trong cae phuang phap danh gia dd tin eay vi no phan anh tinh t y nhien ciia cac yeu td ngau nhien anh hu'dng d i n phan u'ng ciia k i t cau Tuy nhien, nhup'c diem ldn eua phuang phap nay la c i n sd luang l i y m l u ldn, nen neu giai bai toan p h i n tu' hu'u ban vdi so luang p h i n tix ldn va s i l i n phan tich nhieu thi tieu t i n rat nhieu chi phi ve thdi gian va d i u t u cho t h i i t bi phan tich. Nghien cu'u nay dup'c thuc hien de lam k i t qua d i i chimg cho cac nghien cifu t i i p theo eua Chung toi v i danh gia do tin cay cua ket eau cau dan thepa
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Fok. "Reliability assessment pf a typical steel truss bridge," in 7th Austroads Bridge Conference, Auckland, New Zealand, May 2009.
[2]. R. Motamed, S. Elfass and K. Stanton, "LRFD Resistance Factor Calibration for Axially Loaded Drilled Shafts in the Las Vegas Valley," University of Nevada Reno, Reno, NV 89557- 0258, Jul 19, 2016.
[3], J. Huh, A, Haldar, K. Kwak and J.-H. Park, "Realistic risk assessment of axially loaded pile-soil system using a hybrid reliability method," Georisk, vol.
4 , n o , 3 , p p , 118-126, Sep. 2010.
[4]. A. Nowak, "System reliability models for bridge structures,"
Bulletin of the Polish Academy of Sciences, vol. 52, no. 4, pp 321-328,2004.
[5]-A. HaidarandS.Mahadevan, Probability, Reliability and Statistical Methods in Engineering Design, New York:
John Wiley & Sons, 2000.
[6], J. Huh, "Dynamic Reliability Analysis for Nonlinear Structures Using Stochastic Finite Element Method," Doctoral Thesis, The University of Anzona, 1999.
[7], A. Haldar and S. Mahadevan, Reliability Assessment Using Stochastic Finite Element Analysis, New York: John Wiley
& Sons, Inc., 2000.
[8]. AASHTO, AASHTO LRFD Bndge Design Specifications, 4th Edition, Washington, DC:
American Association of State Highv*/ay and Transportation Officials, 2007
[9], TCVN 11823:2017, "Tieu ehuan t h i i t ke cau d u d n g bp,"
Bg Khoa hpc va cdng nghe. Ha Npi, 29/12/2017.
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