KHOAHOC -CONG NGHE ####
cAc no HE^m TIf)]ir TWX
DI; rmn roAxno MES ^vANrmn co m$v
CHO MAT IVTGAJ^G DAM B E TO^G COX THEP
coxfiTDiairjOTrT
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I S . LE BA KHANH Khoa K? thu$t Xdy dyng, Dgi hpc Bach Khoa Tp. H i Chi Minh
KS. TRU'DNG CHJ HUNG Ban Qudn ly Dufl'ng s i t dfl thj Tp. Ho Chi Minh
Tom tdt: Tinh todn dg vong Id mdt trong cdc yeu ciu bit budc ddi vol kit ciu be tdng cdt thdp chju udn 6 trang thdi gioi han S(> di^ng (theo AASHTO). Bg vong cua kit ciu bd tdng cdt thep chju udn ph{j thudc vao dd cCmg chdng udn. Hign nay dnh hu^g cua cdc yeu td nhw vit nui, tCr biin, co ngdt vd lich SLF chit tdi den Dd cOvg chdng udn cOa kit ciu be tdng cdt thdp vin cdn dupc nghidn cOv. Md men qudn tinh cOng vdi md dun ddn hdi dgi didn cho dg cung chdng udn. Bdi bdo ndy gi&i thi$u m'dt sd md hinh m&i di tinh todn md men qudn tinh cd hi$u cua dim be tdng cdt thdp co xdt den nui.
Tif khoa: Md men qudn tinh cd hi$u, Dim be tdng cdt thep, Nui, Vdng, ACI, AASHTO.
Abstract: The limiting of flexural deflection is one of the major serviceability design requirements in reinforced concrete beams. Deflection of reinforced concrete elements depends on the flexural bending stiffness. Currently, the influence of factors such as fracture, creep, shrinkage and loading history to the bending stiffness of the reinforced concrete structure is still being studied. The moment of inertia combined with the modulus of elasticity represents the bending stiffness of a reinforced concrete member This paper introduces some new models to calculate the effective moment of inertia of reinforced concrete beams considering the cracks.
Keywords: Effective moment of inertia. Reinforced concrete beams, Crack, Deflection, ACI, AASHTO.
1.DATVAN0E
Khodng truac ndm 1960 vi$c ktem tra d$ vdng cua kit clu BTCT ehua dupc luu y. Sau thd'i ky nay, cufl'ng dO eua cdc v$t lipu xay dyng nhu BT vd cot thdp dd dup'c cdi thign. Phuang phdp thilt k l theo trgng thdi gifl'i hgn bit d i u dupc dp dgng. Nhfl' dfl ma cdc k i t c l u BTCT trfl" nen thanh mdnh han, dpp han vd tiet ki$m han. Tuy nhidn m|t van de d$t ra d l i vfl'i cde k i t clu thanh mdnh Id phdi dy dodn duyc dO
vong mQt cdch tin cay. Bdi viet sau tim hilu v l cdc mfl hinh tinh mfl ment qudn tinh cfl higu. Ddy la mpt trong cde y l u to chinh dnh hud'ng d i n dO vong eua clu ki$n BTCT
2 NOI DUNG NGHIEN Clirtl 2.1 Gidi thidu v€ tmh toan dp vong
2.1.1 Gidi thifu
Trong nhung nam 1960 nguflfi
ta thdo lu$n nhieu ve gid trj mo bilu dien du6i dgng tong quat:
men qudn tinh ndo dupc diing trong tinh todn dp vflng cua. Trong tho'i gian ndy, dnh huang cua cac ylu t l nhu nui, tu' bien, eo ngflt vd Ijch su chit tai den mfl men qudn tinh chua dupc nghien CCPU day du. Do dfl de dy bao dO vong ngufl'i ta dung mfl men qudn tinh eua tiet dign da bj nCrt d i n tnjc trung hod (J^) hogc mfl men qudn tinh eua tilt di^n nguydn (J^).
2.1.2 Dg vong tifc thoi cua dim Dp vflng ddn h l i cfl t h i duyc
I s l 11 ndm 2014
f # # # # KHOA HQC - C O N G N G H $
A=f/(EJ) (1) Trong dfl:
f - tdi trpng, chieu ddi nhip, dieu kign bidn
EJ - dp cung chong uon;
E - mfl dun ddn hoi;
J - mfl men qudn tinh eua tiet dign.
Doi vfl'i dim giam dan ehju tdi phan bo diu, khi tinh d0 vflng d giua nhjp, hdm f(tai trpng, ehilu ddi nhjp, dilu ki$n bien) cfl dgng
f = 5qL*/384; (2) Trong dfl:
q - tai trpng phdn bo diu;
L - chilu ddi nhjp cua dim;
Neu gid trj tdi trpng ddt Idn clu kign cfln nhfl, khi md Crng suit keo cua bd tflng nhfl han f^. Khi dfl tilt di$n chua bj nCrt, todn b$
tilt dign se tham gia chju lyc. D$
vflng eua tilt di|n:
A = f/lE^-J^ J;
Trong dfl:
(3)
J^ - mfl men qudn tinh quy doi cua tilt dign chua nCrt;
Khi tdi trpng tdng len, nCrt do Cmg suit kdo khi uln xult hi#n.
Mfl men qudn tinh J bj suy gidm vd khfl xdc djnh ehinh xdc. Nhilu nghien cCru chi ra ring khi dfl vin cfl the dung tuang quan sau d l tinh d0 vflng eua clu ki$n (D.E.
Branson, 1977).
A = f/IE^-JJ;
Trong dfl:
(4)
J, - mfl men qudn tinh cfl hi$u cua tilt di$n.
J, cfl gid trj ndm giCa
^9
- r I 1 I 1 1 1 1 1 I VI— 1 1 U U 1 M
Hinh 1. (fng xir dd vdng - tai trgng cua dim BTCT A-BT tgi thd bi keo ddiu dim bit diu bi nirt:
B' BT tgi thd bi keo d giira dim bit dau bj nirt;
C - tai trgng tidu chuin; D &E -cit thdp bi chay.
a} Dd thi thi hign quan h$ dd vdng - tai trgng;
b) Dim vi tai trgng tic di/ng;
A^ v l Jg, duvc coi I I Ithlng dli tren toin bfi chilu dai cua cau kien.
2.1.3. Ouan h$ dd vdng - tai trgng eda dim BTCT
Hinh 1 t h i hien mii quan h | dfi vfing - t l i trpng cua dim BTCT hai diu ngim. Khdi diu, dam chu'a bj nijt (do?n O - A). Khi ting t l i trpng lln, mfi men d diu dam vugt qui nguvng mo men g i y ni>t, khi dfi v l t nCpt do u l n b at diu xult hien.
Khi tilt dien bj nui, mo men quan tinh ciJa tilt di|n b| gilm yeu v l klo theo dp ciing cua d i m b|
giam (do?n A - B). V l t raH xult hien d giira nhjp tiep tuc Ilm giam dp ci>ng (dilm B). Culi ciing, cit thep cfi t h i bj chay d vung dau dam hole giu'a d i m (dilm D). Khi dfi tai trpng ting it nhung biln d^ing ting rat nhilu. Trong giai doan khai thic, i>ng v*i tai trpng tilu chuin, d i m phli cfi img xii' d i n hoi (diem C). Nlu tai trpng
mm
tilu chuin du'pc duy tri liu dai, dfi vfing cua dam cfi t h i tang tir diem C den dilm C
2.2 Mo men quan tinh cd hieu 2.2.1 Md hinh eda Branson N i m 1963, Dan E. Branson dl cflng bo nghien ciru ding chii y vS dfl vfing cua c l u ki|n BTCT khfing u'ng suit tru'fi'c (Branson 1963). Branson d l kiem tra tinh dung d i n cua mfl hinh bing cicli so sinh dfl vong cOa clu ki|n c6 tilt difln chO' nh|t v l chu' T chju tai trpng phin b l dSu.
J. = -la + (-',-J„)(M„/M.)'; (5) Trpng do
M» = ngu'fi'ng mfi men giy ni>l
J, - mfl men quin tinh cua tilt difin BTCT chu'a nirt (nguyen);
1, - cu-fing dfi chju k i p khi uln cua BT thyfi'ng = 0,63(f y.^-
y, - khoang cich tir trpng tim
S4l1 r
KHOA HQC-CONG NGHE t i t t .
1
1— °I=L
' — B
^
Hinh 2. Si/ biin thidn dd cung dge theo chiiu ddi dim
din thfl' chiu kdo ngodi cung;
M, - mfl men uon tieu chuin lan nhat fl' thfl'i dilm chat tdi.
Tren ca sa kit qud thi nghidm, Branson d l xult: dli vai vung cfl gid trj mfl men khflng dli a = 4, dli vdi dim gidn dan a = 3. Bilu thuc tinh J^ cua Branson vfl'i a = 3 dupc ACI chip nhdn dua vdo tieu chuin tCr ndm 1971 cho d i n nay (ACI 318-14). Bd qua dnh hud'ng cua d t thdp, ACI cho phdp dung tilt di§n BT de tinh J^.
2.2.2 Anh huvmg cua sc d6 chit tdi den mo hinh mo men quin tinh cd hifu
Vdo ndm 1991, cdc hpc gid tu' Dgi hpc King Saud d Riyadh, A Rdp Saudi cong bd nhung phdt hi^n tCr nghien ciru md hp tiln hdnh d l xdc djnh dnh hud'ng eua sa d l chit tai d i n mfl hinh mo men qudn tinh cfl hi^u cua Branson (Al-Zaid Al-Shaikh, vd Abu-Hussein 1991).
Cdc gid tri thye nghidm dd Chung minh rang mfl hinh Branson khflng the chinh xdc cho tit cd cdc trufl'ng hp'p tai. Theo kit qua thi nghi|m mfl men qudn tinh thye ir sa d l chit tdi t$p tnjng tgi giu'a nhjp cfl the lfl'n 20%
so vfl'i sa d l chit tdi phdn b l deu.
2.2.3 Anh hu'dng cua hdm Itfgng c6t thdp din m6 hinh md men qudn tinh cfl hifu
Nhflm tdc gia tren eung da nghidn eCru ede dim bd tflng cit thdp vfl'i hdm lupng cot thep thay dli. Cde dim khdo sat efl mgt c i t ngang hinh chu nhgt, chju tdi trpng tgp trung tgi giua nhjp.
Nghien eCru da chi ra ring gid tri mfl men qudn tinh co higu theo Branson cfl t h i nhfl han khodng 12% - 30% so vfl'i thye t l . Dde bigt khi hdm luyng cot thdp cdng nhilu, khae bipt ndy edng lan.
2.2.4 Mo hinh mdi cua A.M.
Fikry vd C. Thomas Ndm 1998 nhflm nghien cuu A.M. Fikry vd C. Thomas da cfing b l m$t nghidn cuu ve mo hinh tinh mfi men qudn tinh cfl hi§u dya trdn ea sfl* ly thuyit ve Crng xu cua clu ki$n ehju uln BTCT.
Cdc tdc gia da logi bfl vi^c tinh todn J^^ vd ndng cao tinh hpp ly bing edeh dua thdm cdc tham so d l xet d i n hdm lupng c i t thdp vd sa do chat tdi. Sai s l cua mfl hinh so vfl'i kit qua thi nghigm nim trong khoang 6%. Mfl men qudn tinh efl higu theo A.M. Fikry vd C.
Thomas dupe xdc djnh doi v6i d i m BTCT tilt dign ehu nhgt, dgt cit dan:
J „ = (a + Pnp)-(bd712) (6) Trong dfl:
0, P = cdc hing s l ; n = t^ s6 mfl dun ddn hli cua
thdp va BT = EJE^.
p = hdm luyng eit thep;
b = chilu r|ng cua cau kign;
d = chilu cao Idm vi|c cua dim BTCT
2.3 Vl du tmh J , theo mo hlnh Branson
Cho mflt Dim BTCT tilt di§n chu nhgt, dgt cot dan efl eac thong so nhu sau
Sang 1. So Hgu ve dam BTCT
b mm 200
h mm 400
L mm 4600
V
MPa 28
f j
MPa 420
E, MPa 200 000
11
As
Hinh 3. Ciu tgo m$t cit ngang dam BTCT Diing hai cly thip 028; Lfi>p BT bio ve 25 mm;
Tinh dfl vfing tgi m|t c i t giOa nhjp ciia dim. Tli trpng t i c dung lln dim gom trpng (uvng ban thin dam v l hogt tai lln xe 9,3 N/mm;
Gi'ai
Dfl vong ifi^n nhit duvc xic djnh theo cfing thirc
A = 5qL'/384/(EJ)
Tinh f, - cu'fi'ng dfl chju keo khi uon cua BT thu'firng = 0,63(f,,')°';
f, = 3,3336 MPa: . Tinh J^ - mfi men quan tinh cua tilt di#n BTCT chu'a nirt;
ACI cho phip diing tilt nguyen
I s i 11 nam 2014
r # # # # KHOA H O C - C O N G NGHE
khfing x l t d i n d t thip;
J =bh>;i2;
B
J = 2 211 840 000 mm';
0
Tinh M„ - ngirang mfi men gay nirt;
M = 17 779 449Nmm;
Tinh p - him lu'png cot thep;
p = 0,0171
Tinh n - ty s l mfi dun din hli ciia thip v l BT = E./E,,;
n = 7,4759;
Difn tich d t thip chiu lyc A,= 1231,5 mm=;
T l n h J , = b/3(kd.)' + n A , ( d . - k d / ;
J,^= 632 443 768 mm';
Du'fiu t i c dgng cua tmh tai tilu chuin v l hoat tai xe tilu chuin, dim v i n Ilm vipc d giai doan dan hli.
Mo men quin tinh d hi|u
•'. = -'»*(-'g"-'J('*''»'^.''' M^ - mfl men uln tieu chuin I6n nhit d thcri dilm chit tai.
Chi do tinh t l i dim M, 5 760 000 Nmm < M^
(dim chu'a bj ni>t);
Bing 2. Kit qui tinb vong
Chi do hogt tli l l n xe M = 26 784 000 Nmm > M , (dim da bj nirt);
Theo bing 2. khflng the nfli rang dfl vflng do hogt tli bing 3,16 mm. Khi chit thim hogt tli lln dim, dim phli chju ca tmh tai v l hogt t l i , do dfi mirc dfl nirt se nhilu hon khi chi d|t mflt minh hogt tai. Do dfl dfi vflng do hogt tai:
4,15 mm - 0,48 mm = 3,67 mm;
3.KETLU&N
Mgc dil nhieu nghien ciru d l chi ra nhOng thilu sfit cua mfl hinh Branson (bilu thirc 5). Hifln nay (din nam 2014), mo hinh ciia Branson v i n dup'c irng dung trong hau hit cic tieu chuin ciia My d l thilt ke kit d u b l tfing d t thip (tieu chuin ACI 318-14 - 24.2.3.5, tilu chuin AASHTO - 5.7.3.6.2).
Dfi chinh xac cOa mfi hinh Branson van con la v i n d l , dac bi|t khi chit t l i t|p trung. Day I I vin de d n lu'u y khi kilm tra vflng ciia d u dim BTCT chju hogt tai xe thilt k l .
Khi tinh vfing ciia dam BTCT nfii Chung va d u dim BTCT nfii riing, do sy xuat hign v l t nirt cho nen khfing t h i I p dgng nguyen t i c cflng t i c dung.
Til trpng TlnhtSi Hoattji TInntii>ho«ttii
q N/mm
2 9,3 11,3
II.
N-mm 5 760 000 26 784 000 32 544 OOC
J.
mill*
1 066 666 667 759 454 755 703 247 281
A mm
0.48 3,16 4,15
TAI UEU THAM KHAO
;1 American Concrete Institute (ACI) 2014 'Building code requirements for stmctural concrete and commentary', ACI 318-14, ACI Committee 318, Famiington Hills, Mich.
2. AASHTO LRFD Bridge Design Specifications 2012 ed., American Association of State Highway and Transportation Officials, Washington, D.C., 2012 3. Joseph E. Wickline "A
Study of Effective Moment of Inertia Models for Full- Scale Reinforced Conctet*
T-Beams Subjected to A Tandem-Axle Load Configuration", M.Sc.
Thesis, Dept. of Civil and Envir. Eng. Faculty of the Virginia Polytechnic Institute and State University.
4. Fikry, A.M. and Thomas, C., "Development of a Model for the Effective Moment of Inertia of One- way Reinforced Concrete Elements," ACI Structural Joumal, V. 95, No. 4, July- August 1998, pp. 444-455 5. Nilson, A.H., Danwin, D. and
Dolan, CW. 2010, Design of Concrete Structures, 14tli edition, McGraw-H ill, Boston 6. Wight, J.K, Macgregor, J.G.
2012, Reinforced Concrete:
Mechanics and Design, 6th edition, Pearson, Boston.
