Cong trinh Khoa hgc
NGHIEN ClTU ITNG DUNG HAM SOLVER XAC DINH BtronVG DO TOI iTU TRONG THIET KE TRAC DOC TUYEN DU'CJfNG
LUONG XUAN BINH HOANG THANH THUY Tnrdng 0gi hoc Giao t/iSng Vgn tdi D A O N G Q C HilNG
S& Giao thdng Vgn tai Yen Bdi
Tdm till: Cong ldc thiel ki trdc dgc cua tuyin dudng sao cho gidm thieu tSi da khoi litgng ddo ddp cUng nhu cht phi xdy dung trong khi vdn phdi ddp irng dirge nhiing yeu cdu nghiem ngdt vi cdc liiu chudn ky Ihudi Id cong viic rdt phuc tgp, kho khdn Ldm the ndo di xdc dinh duac dudng dd loi uu vdi hdm muc lieu toi uni ddng thdi thda man cdc rdng huge ve tieu chudn ky thugt co mgt y nghTa rdt quan trgng trong thiet ke Irdc doc tuyen dudng. Bdi bdo ndy gidi thieu mot nghien ciru ung dung noi hdm Solver trong Microsoft Excel de gidt quyet bdi todn noi tren
Summary: Design work of vertical alignment of roads, of which the volume of cut-fill or construclion costs are optimized with the satisfaction of very strict requirements on technical specifications, is quite complicated. How lo determine the appropriate vertical alignment with an optimal target and satisfaction of restraints on technical specifications has a remarkable meaning in design of roads. This paper deals with the research on an application oj Solver Function in Microsoft Excel to resolve the problem mentioned above.
I. D A T V A N D E
Chpn vj tri d u d n g d o (mat cat dpc thiet ke) toi uu la bai toan kinh te t o n g h g p xay dyng - khai thac van doanh. Doi vdi vung d i i niii, d u d n g d o t h u d n g d u g c thiet ke theo p h u a n g phap d u d n g cat, d dd d u d n g dd di cat d u d n g mat dat t y nhien va t^o thanh nhitng doan d u d n g dao dap xen ke, Khi nay, vifc xac dinh d u d n g d o trong t h i l t ke trae d g c t u y i n d u d n g t h u d n g dupc thuc hifn theo p h u a n g p h a p lap di lap lai nhieu lan, m u c dich de thoa man tat ca cac dieu kien v l thi cong, khdi l u g n g dao d i p , t r i e dpc, trae ngang, tam nhin, dp d i e dgc, doan doi d i e , t i n g miic dau t u va cac d i l u kien khac theo tieu chuan t h i l t ke d u d n g d u g c phep ap d g n g [1]. Cong viec do ton rat nhieu thdi gian va cong sire ctia ngudi ky su.
Nhieu tac gia da di vao nghien cuu van d e nay va da d e xuat nhieu giai phap Irong nhung thap nien vira qua. Tinh toan xac dinh d u d n g do d u g c thyc hifn t h e o hai giai doan: Giai doan 1 - Xac djnh d u d n g dd kha thi; Giai doan 2 - Tdi uu hoa khoi l u g n g dao dap. Nghien cuu d i u tien da k i t hop hai giai doan noi tren trong mpt m o hinh tinh co t h i tim t h i y trong [2] va [3]. N h i l u p h u a n g phap xac djnh d u d n g d o thong d u n g c o thf ke den n h u : p h u o n g phap tim k i l m so, p h u a n g phap quy hoach dpng, p h u a n g phap quy hogch tuyen tinh, p h u a n g p h a p q u y hogch toan
70 Tap chi KHOA HQC GIAO THONG VAN TAI S6 47 - 08/2015
Cong trinh Khoa hgc phuang...
NhCrng phuang phap tim kilm s l dau tien su dung khdng gian tim kilm lien tuc, d do nhijng thanh phan chi phi va cac dieu kien rang bugc dugc xac dinh mdt each truc tiep, tuy nhien phuang phap nay din tdi viec phai giai mpt bai toan phirc tap do tinh phi tuyen va khong hpi tu ciia mo hinh tinh. Hayman [4] da sir dung nhung bien lien tuc md ta cao do tai cac mat cit va sir dung thuat toan tim kifm sd de xac djnh cac cao dp tdi uu va thda man cac dieu kien rang bupe cua bai toan. Dudng do la dudng gap khiic noi cac diem cao dp dugc xac dinh noi tren.
Phuang phap nay co mot nhuac diem la chi co the xac djnh dugc dudng dd tdi uu trong difu kien dudng dd gia djnh ban dau kha gan vdi dudng do toi uu can xac dinh. Phong nghien ciru ve cong trinh dudng va giao thong van tai (TRRL) d Anh quoc da xay dyng mgt chuang trinh may tinh MINERVA [5, 6], trong do tac gia cung sir dyng phuang phap tim kiem so de xac djnh dudng do, tuy nhien khac vdi Phuang phap Hayman, thay vi dudng do la dudng gap khiic thi d day dudng do la dudng to hgp bdi nhDng doan thang va cac doan cong parabol Phuang phap nay ciing phu thudc nhieu vao dii lieu dudng dd gia dinh ban dau cd gan vdi dudng do toi uu hay khong. Robinson [7] da xay dung mot chuang trinh VENUS de xac dinh dudng do bang each bien doi dudng cong mat dat thien nhien thanh dudng do bang phuang phap binh phuong toi thieu. Phuang phap nay khong phai liic nao cung co the xac djnh dupc ket qua toi uu toan cue. Goh va cac dong nghiep [8] da xay dyng mgt mo hinh lien tuc va mot mo hinh rdi rac de toi uu hoa dudng do. Trong md hinh lien tuc, Goh da lan dau tien xay dyng ly thuyet tinh toan xac djnh dudng do nhu la mgt bai toan bien phan, sau do chuyen doi sang bai toan dieu khien toi uu [9]. Tuy nhien, viec giai bai toan la rat phuc tap va mpt so trudng hgp khong ap dung
• dugc phuong phap nay.
Phuang phap quy hoach dgng lan dau tien dugc irng dung de giai bai toan toi uu hoa dudng do la do Puy Huarte [10]. Tac gia da phat trien mo hinh hai giai doan, d do, giai dogn Ihir nhat, Iy thuyit quy hogch dgng da dugc ap dung de xac dinh dudng do tdi uu d dgng dudng thang giy khiic, sau do, dudng cong bac ba dugc sir dung de xap xi hoa dudng dd trong giai dogn 2. Goh vacac cgng sy [8], Fwa [11], va Goktepe vacac cpng su [12] d§ sir dyng mot con dudng tuang ty dl toi uu hoa dudng do bing phuang phap quy hoach dgng. Cac tac gia da rdi rac hoa khong gian tim kiem bang cac dudng ludi nam ngang va thang diing. Dudng do dugc toi \.n\ theo mgt qua trinh nhieu giai doan dgc theo cac dudng ludi chia.
Quy hogch tuyen ti'nh cung la mgt phuong phap kha Ihong dung. Mpt chuong trinh luyen tinh vdi mgt ham muc tieu tuyen tinh va cac rang bupe tuyen tinh doi vdi cac bien xac dinh cua bai toan. Y tudng sir dung phuong phap quy hoach tuyen tinh vao toi uu hoa cong tac dao dap dugc ban luan lin dau tien trong [13]. Sau do, Mayer va Stark [14] phat triln mo hinh cue tieu hoa chi phi van chuyen dat da,
d Vift nam, Pham Van Ky [15] da phat triln mgt mo hinh toi uu hoa trie dpc dai tu dudng sit bing phuang phap quy hoach dpng. Gin day, Heng Kim Leng da tien hanh nghien ciru de tat
"Nghien ciru thilt ke binh d l trie dpc hgp ly dudng 6 to" [16]. Trong do, dudng do dugc xac dhih bing phuang phap lap khi thay dii tpa do cac dinh goc chuyln hudng ciia dudng do theo
Tap chl KHOA HQC GIAO THONG VAN TAI S6 47 - 08/2015 71
ludi tpa do chia san. Vdi phuang phap nay, khii lugng tinh toan Idn va tdn nhilu thdi gian de xac djnh dugc duong do toi uu.
Lam sao de dua ra dugc phuang an tdi uu boa dudng do trong thilt kl trie dgc tuyen dudng, d do tiet kiem dugc thdi gian, cong sire ciia ngudi thilt kl la vifc cd y nghTa het sire can thilt. Trong phgm vi bai bao nay, cac tac gia tgp trung nghien ciiu xay dung mdt phuang phap hieu qua giai bai toan toi uu de xac djnh dudng do tuyen duong thilt kl bang each iing dung mpt npi ham cua Microsoft Excel - Ham Solver. De dan gian boa bai toan, trong phgm vi bai bao nay, bo qua cac dudng cong dung tren trae dgc ma khdng anh hudng nhieu den ket qua toan cue ciia bai toan.
11. CO SO LY THUYET BAI TOAN TOI UU HOA XAC DINH VI TRI DUONG DO TRONG THIET KE TRAC DOC TUYEN D U Q N G
Hdm muc tieu ciia bai toan ky hieu la f(X). Ham muc tieu nay cd the la: Khoi lugng dao khi thi cong tuyen, khoi lugng dap khi thi cong tuyen, tdng khoi lugng dao dap khi thi cong tuyen. hoac tong chi phi dao dap khi thi cong tuyen... Viec chgn ham mue tieu nao trong nhung dgi lugng ke tren la phu thugc vao dac diem yeu eau ciia timg dy an cong trinh.
Tham so bien cua bai toan X = {xi, Xi, Xj,..., Xn}, la cac bien sd cd gia tri thay doi trong suot qua trinh toi uu hda. Trong do, moi thanh phan tham so bien (x,) la mgt dgc trung hinh hpc CO gia tri bien ddi trong qua trinh tinh toan cua dudng do tuyen dudng thiet ke. Cd the ke den cac tham so bien nhu sau: Chieu dai doan ddi ddc (chieu dai canh tuyen), dp doc dpc ciia moi canh tuyen...
Cac dieu kien rang bugc cua bai toan bao gom: Cao do ddu tiiyen: Phu thugc vao dja hinh, cong trinh thoat nudc, cac diem khdng che theo quy hogch, cong trinh cong cgng co san, va cac
>eii to khac... Cao do cuoi tuyen: Tuong ty nhu ddi vdi cao do dau tuyen. £>g doc dgc tuyen:
Tuan thil theo tieu chuan thiet ke dugc ap dung. Chieu ddi doi doc: Tuan thii theo tieu chuan thiet ke dugc ap dung. Diim khong chi: Tuan thii theo cac dieu kifn ve quy hoach, cac cong trinh cd san khdng thf di ddi, thay doi, cac cdng trinh thoat nudc. Do doc ngang mat dirang:
Tuan thii theo tieu chuan thiet ke dugc ap dung. Tdm nhin: Tuan thii theo tieu chuan thiet ke dugc ap dung. Bdn kinh dirang cong nam: Tuan thii theo tieu chuan thiet kf dugc ap dung. Bdn kinh dircmg cong dimg: Tuan thii theo tieu chuan thiet ke dugc ap dung. Dieu kiin vi von: Can cir vao tinh hinh thyc te, von hien co de dau tu xay dung cdng trinh, ke hogch vdn giao. Diiu kiin ve mo ddt ddp, bdi do ddt: Trii lugng va cy ly van chuyen md dat dip, bai dd dat dao va cy Iy van chuyen, don gia van chuyen. Vd cdc dieu kiin rdng bugc khdc: Ngudi tinh co the bo sung cac dieu kifn rang bupe phii hpp vdi dieu kien thyc te dy an dang xet...
Phdt biiu bai toan: Xac djnh gia trj tham so bien X^= {x*,X2,x,,...x^} sao cho ham muc tieu f{X) dat gia trj min hoac gia trj max ding thdi tat ca cac dieu kien rang bupe ciia bai toan can phai dupc thoa man. Cac gia trj tham so biln khi nay cung chinh la cac dac trung hinh hgc xac djnh dudng do hgp ly ciia bai toan.
72 Tgp chi KHOA HQC GIAO THONG V ^ N TAI S6 47 - 08/2015
Cong trinh Khoa hgc III. iTNG DUNG HAM SOLVER DE GIAI BAI TOAN TOI UU HOA XAC DINH DUOT^G DO TRONG THIET KE TUYEN DlTOfNG
3.L Gioi thieu ham Solver
Solver la mgt trong nhirng npi ham ciia Microft Excel, dugc xay dyng va dua vao su dyng tir phien ban Microft: Excel 97. Vdi Solver, ngudi dimg cd the giai cac bai toan sau day thong qua bang tinh Excel: Giai he phuang trinh tuyfn tinh, phuang trinh phi tuyen, cac phuang trinh dai so bac cao, sieu viet, ham mii, giai cac bai toan toi uu.
3.2. Ap dung ham Solver giai bai toan toi uu hoa dudng do
Chgn 6 dich can ti'nh bang gia trj ham muc tieu. Chgn 6 can tinh Solver, chgn gia trj cin tinh "Min" hoac "Max", hoac gia trj "Value of. Biln sd: Ta chgn cac bien tren bang tinh la cac dp doc dpc cua dudng, khoang each giira dilm dii ddc da co va cpc ty nhien trudc ciia no roi dgt vao 6 "By Changing cells". Theo difu kifn dau bai, ta chpn cac dieu kien rang bupe tren bang tinh la dp doc dpc i%, chieu dai doan doi doc L, cao dp khong chf, khoi lugng dao, dap, dy toan kinh phi... rdi dat vao d "Subject to the Constraints". Them cae dieu kien vao 6 "Add", hoac thay dii "Change" hoac xoa "Delete". Vao "Options" de cai dat cac thong sl cin thilt trong hop thogi "Solver Options". Tien hanh chay Solver, tren bang so lieu trie dgc, trie, ngang se thay doi cac gia tri theo cac dieu kien rang bupe tgi 6 "Subject to the Constraints".
Cu6i cimg Solver cho ket qua la 6 gia tri can tim va Excel se cho bieu do tuang irng vdi ket qua dgt dugc Trong trudng hgp vdi bai toan lap Tdng mire dau tu, xac djnh Tdng mire dau tu toi uu, thi ta tien hanh Igp Dy toan cong trinh theo cac hudng dan cua Bg xay dyng hien hanh. Tir do dimg Solver df tinh toan loi uu cho Tdng muc dau tu tucmg ty nhu phan tinh toi uu khdi lugng dao dap
IV. VI DU TINH TOAN VA DANH GIA KET QUA 4.1. Thi du dan gian, mo ta
ham muc lieu kiem chiirng ket qua tinh toan
Buv'c 1: Nhap sd lifu dau vao (bang I). Diem khong chl: Cao dp dau tuyen triing vdi cao do ty nhien, cao dp cu6i tuyen la: 50m.
Bw&c 2: Nhap sd lifu cac mgt cat ngang (sd lifu dudng den va dudng do gia djnh ban dau).
Biv&c 3: Nhgp so lifu tren bang mat cit dpc (so lifu dudng den va dudng do gia djnh ban dau).
Biedc 4: Dimg ham Solver tinh toan.
Tru-dng hap 1: Ham muc tieu f(X) = [Khii lugng dip (duong) + Khoi lugng dao (am)] -> min.
Kit qua tinh: Khoi lugng dao la: 950, Itm^ bing vdi khdi lugng dip, tong chi phi dao dap la 71.223.524 dong. Dudng do tii uu dugc thi hien tren hinh 1.
Tnrdng hop 2: Ham mgc tieu f(X) = Khii lugng dip (cl dinh) + Khii lugng dao (nho nhit).
TT 1 2 3 4 5 6 7
Bdng I. Sd lieu ddu Hang muc Cap dudng thiet ke Tieu chuan thiet ke Ap dung
Do doc ngang m$t dudng Be rong mat dudng Do doc mai ta luy Chilu dai dli d i e t i i thieu Do doc doc toi da
vdo vi du linh todn dan gidn Tieu chuan Cap 111, mien
niii TCVN 4054-
2005 4 20 1/1 300 11
Don vi
%
mm
%
Tap chi KHOA HQC GIAO THONG VAN TAI SS 47 - 08/2015 73
Kit qud linh: Khdi lupng dao la: 2000m^ khii lugng dip la 498,16m\ tong chi phi dao dip la 69.643.628 dong.
Dudng dd tdi uu dugc the hifn tren hinh 2.
Trir&ng hgp 3: Ham muc tieu f(X) = Ting chi phi dao dap (nho nhit).
Ket qud linh: Khii lugng dao la: I.491,51m\ khoi lugng dip la 623,79m\ tong chi phi dao dip la 65.779.091 dong. Dudng do tdi uu dugc the hifn tren hinh 3.
Nhgn xet: Cac trucmg hgp tinh toan dan gian cho thay hinh ve trie dgc tuyin dudng cho khoi lugng dao dap phii hgp vdi ham muc tifu tuong iing, difu dd cho thay kit qua,tinh toan la dang tin cgy. Ngoai ra, khi so sanh ba trudng hgp d trfn, trudng hgp 3 cho ting chi phi dao dip nho nhit.
4.2. Ap dung tinh toan cho cdng trinh duwng Hoa Cuong - Tan Hirong, huyen Tran Yen, tinh Yen Bai
Birac 1: Nhap sd lieu dau vao (bang 2). Dilm khdng chl: Cao dp diu tuyfn trung cao dp ty nhien, cao dg culi tuyfn la: 60m. Cac thdng tin khac: Bai do dat cho phep do thi d l tii da 10.000 m \ cd khoang each vdi diu tuyen dudng la 5Km; Mo dat dap co triJ lugng khoang 12.000m\
CO vj tri each dau tuyen 7Km;
Dit dao cd the tan dung 80%
de dap.
Bdng 2. Sd lieu ddu v n du linh lodn dan gidn Hang muc
Cap dudng thiet ke Tieu chuin thiel ke ap dung Dp doc ngang mat dudng Be rong mat dudng Do doc mai ta luy Chieu dai doi doc toi thieu Do doc doc toi da
Tieu chuan Cap III, mien TC VN 4054-
2005
Dan vi
TBAC D Q C TliYtN BirdNC
150 700 7y) 3D0 150 400 ^00 SSO Geo 6 U
Hinh 1. Kit qud xdc dinh dudng_ do hap ly sao cho khoi lugng dao bdng khoi lugng ddp
TRAC D p c TllVtN OirdNG
Hinh 2. Kit qud xdc dinh dudng do hgp ly sao cho khdi lugng dap co dinh, khdi lugng ddo nho nhdt
'Z •.... "-
wn .
y^
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TRAC DOC TUYtN DlfdNG
Mai dll
• r —
JtO ? » 300 ( n
] Duonu do hqp ly i
/i
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I'iD 400 < i n MD 5in 600 (,'iO
\
Hinh 3. Kit qud xdc dinh dudng do hap ly sao cho chi phi ddo dap nho nhdt
74 Tap chi KHOA HQC GIAO T H 6 N G VAN TAI S6 47 - 08/2015
Cong trinh Khoa hgc Budc 2: Nhap s l lieu cac mat cit ngang (sl lieu dudng den va dudng do gia dinh ban dau).
Budc 3: Nhap sl lieu tren bang mat cit dpe (sl lifu dudng den va dudng do gia dinh ban diu).
Birdc 4: Dimg ham Solver tinh toan. Ham myc tieu f(X) = Tdng chi phi dao dap (nhd nhat).
Kit qua tinh: Khii lugng dao la: 15.244,36m\ khii lugng dap la 20.l75,59m\ tdng chi phi dao dip la 1.330.386.241 ding. Dudng dd til uu duge thf hifn tren hinh 4.
Hinh 4. Kit qud xdc dinh dudng do hgp ly tuyen dudng Hoa Cuong - Tdn Huang, tmh Yen Bdi V. K E T L U A N
Nghien ciru da ung dyng thanh cong Ham Solver, mot npi ham ciia Microsoft Excel, vao xay dyng mdt phuang phap toi uu boa xac dinh dudng do tdi uu trong thift kf trae dpe tuyen dudng. Phuang phap nay cho ket qua tinh toan dang tin cay, cho phep kJ su thiet ke co the xac djnh dugc dudng dd hgp ly theo muc-tieu thiet ke mgt each nhanh chdng va thuan tien. Bfn cgnh do, phuang phap nay ciing cho phep ngudi thiet ke c6 the xac djnh dudng do theo nhieu ham muc tifu khac nhau, tir do cd the so sanh de lya chgn dupc phuang an dudng do phii hgp nhat.
Ng^y nh^n bai lan dau: 14/5/2015 Ng^y nhan bai sua: 2/6/2015 Ng^y chap nhan dang bai: 11/6/2015
T&i lifu tham khao
[I]. Nguyin Xudn True. Duong Hgc Hai. Vu Dinh Phung: S l tay thilt kl dir&ng 6 to - Tap 1, Nh^ xuat ban Gido due. 2001.
[2]. A. B. Baker. The design and phasing of horizontal and vertical alignments: Program JANUS. Transport and Road Research Laboratory, 1972.
[3], Ontario Ministry of Transportation and Communications. GeomeUic design standards for Ontario highways. Information Management Office, Downsview, Ontario, Canada, 1984.
[4]. RWHayman. Optimization of vertical alignment for highways through mathematical programming.
Highway Research Record, pages 1-9, 1970.
[5]. OECD. Optimisation of road alignment by the use of computers. Technical report, Organization of Economic Co-operation and Development, Paris, 1973.
[6]. A D Pearman. Investigation ofthe objective function in problems of optimising highway vertical alignment. In PTRC Seminar Proc, Cost Models and Optimisation in Highways, Session LIS, London, 1973,
Tap chi KHOA HOC GIAO THONG V ^ N TAI So 47 - 08/2015 75
[7]. R Robinson. Automatic design of the road vertical alignment. In PTRC Seminar Proceedings on Cost Models and Optimization in Highways, Session L9, London, UK, 1973.
[8]. CJ. Goh, EP. Chew, and TF Fwa. Discrete and continuous models for computation of optimal vertical highway alignment. Transportation Research Part B: Methodological, 22(6):399 - 409, 1988.
[9]. C / Goh and K. L Tea. Control parametrization: a unified approach to optimal control problems with general constraints. Automatica, 24:3-18, January 1988.
[10]. J Puy Huarte Opygar. Optimization and automatic design of h i ^ w a y profiles. In PTRC Seminar Proceedings on Cost Models and Optimization in Highways, Session LI3, London, UK, 1973.
[II]. r F. Fwa. Highway vertical alignment analysis by dynamic programming. Transportation Research Record, 1239:1-9, 1989.
[12]. A. Biirak Goktepe, A Hilmi Lav. andSelim Altiin. Dynamic optimization algorithm for vertical alignment of highways. Mathematical and Computational Applications, 10(3):341-350,2005.
[13]. Robert M Stark and Robert L Nicholls. Mathematical Foundations for Design: Civil Engineering Syslems.
Technology and Engineering. McGraw-Hill, 1971.
[14]. S. Nandgaonkar. Earthwork transportation allocations. Journal of Construction Division, 107:373-392, 1981.
[15]. Pham Vdn Ky: 'Thiet kl tii iru trie doc dai tu 6utmg sit", Lu|n an Pho tien sT, Truong Dai hpc GTVT, 1995, [16]. Hing Kim Leng: "Nghien cihi thiet ke binh do trie dpc hgp ly duong 6 to", Luan an ti^n sT k^ thuyit, Truang Dai hoc Giao thong V ^ tai, 2006*
Phan bien: VQ fitlc S^; Nguyen Quang Phuc
THIET KE VilNG CHIU L V C CUC BO ...
(Tiep theo irang 64) IV. K E T L U A N
Bai'bao da trinh bay ket qua tinh toan lugng cot thep chiu luc can thiet cho mgt kft cau be cgc khdng gian dien hinh trong eac trudng hgp ehju lyc nen va khoang each cgc khac nhau theo phuang phap MHHT cua tifu chuan 22TCN-272-05 so sanh vdi phuang phap mat cat phang thudng dimg trong cac viing chiu uon thong thudng. Ket qua cho thay phuang phap mo hinh he thanh luon cho gia tn elt thep thien ve an toan ban trong cac trudng hgp da xet va han n&a la chi ra nhOng vi tn can bo tri cot thep ehju lyc chinh de ehju cac lyc keo sinh ra trong be cpc.
Dieu nay khang djnh ti'nh hiju ich cua phuang phap MHHT khi thiet ke cac vung cue bp luong ty trong ket eau BTCT.
Ngay nhan bai |4D dau: 11/5/2015 Ngay nhan bai sua: 2/6/2015 Ngay chap nhSn dang bai: 11/6/2015
Tai lieu tham khao
[I]. American Association of State Highway and Transportation Officials: AASHTO LRFD Bridge Design Specifications., SI Units, Second Edition, 1998.
[2]. ACI Committee 318: ACI 318-02: Building Code Requirements for Structural Concrete {ACI 318M-02) and Commentary (ACI 318RM-02).
[3]. CEB-FIB Mode! Code 1990: Bulletin D'Information N= 203 und 204, July 1991.
[4], Canadian Standards Association: CSA A23.3-94: Design of Concrete Structures, 1994.
[5]. DIN 1045-1: Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1; Bemessung und Konstruktion, Juli 2001 [6]. ENV 1992-1-1: Eurocode 2 Planung von Stahlbeton-und Spannbetontragwerken. Teil I: Grundlagen und Anwendungsregeln fiirden Hochbau, Oktober 1991.
[7]. Schlaich. J. Schafer. K.: Konstruieren im Stahlbeton bau. Betonkalender 2001 Teil II, Ernst und Sohn Verlag, Berlin 2001.
[8] Bo Giao thong vSn tai" Tieu chuin thilt k l Cau 22TCN 272 - 0 5 * Phan bien: Tran Thi Truyln; NgS VSn Minh
76 Tap chi KHOA HOC GIAO THONG V^N TAI So 47 - 08/2015
