T I N H ON D I N H THANH CHIU NEN C O LIEN K E T DI HlTOfNG
LUONG XUAIN BINH fiOXUANQUY Bp mdn Stic ben vat lieu Trudng Bai hoc Giao thdng Van tdi
Tom tdt: Bdi bdo giai thieu mgt phuang phdp tinh on dinh thanh chiu nen cd lien kit dj hudng. Trong phuang phdp ndy, tdc gid dd khdc phiic sie phuc lap loan hgc trong bdi loan on dinh bdng cdch dgt vdn de gidi bat todn theo phuang phdp nica nguac Dang duong dan hoi ciia thanh lai trgng thdi giai han duac gid dinh dudi dgng mdt da thuc bgc cao. Cdc he sd ciia da thirc nay dugc xdc dinh bdng cdch giai bdi loan tdi uu hda hdm muc lieu kht phuang trinh ddn duong ddn hdi cua thanh phdi thda mdn phuang Irinh vi phdn ca ban Irong bdi loan dn djnh, ddng thai thda man cdc dieu kien bien cua thanh Khi dudng ddn hdi duac xdc dinh, luc tdi hgn cua thanh dugc xdc dinh mot cdch don gian bdng cdng thirc ndng lirong. Bdi bdo ciing trinh bay cdch xdy dung thuat todn. chuang trinh tinh tren mdy tinh.
Abstract: This paper deals with a method Jor analysis of stability of a bar subjected to compression with anisotropic restraints. In this method, the author overcomes the mathematical complicatedness m stability problems with using semi-inverse solutions. Elastic curve of the bar at critical state is assumed under the form of a multinomial of which the parameters are defined with optimizing objective function when the multinomial is satisfied the fundamental differential equation of the .stability problem and the boundary conditions of the bar. When the elastic curve is defined, the critical force is simply defined with energy formula.
The paper also demonstrates the establishment of algorithm and computer program lo solve the problems.
L f i A T V A N D E
Trong ky thuat ndi c h u n g , ngdnh xay d u n g c d n g trinh ndi rieng, viec phdi xet den dn dinh ciia cdc cau kien chiu nen kha phd bien vi dy n h u dn dinh ciia try cau, thdp cdu treo day vang, ray trong d u d n g sat... T r o n g dd lien ket tai cac dau cau kien la nhffng lien ket phirc tap nhu: lien ket Cling, lien ket ddn hdi, lien ket day, lien ket phi t u y e n . . . T r u d n g h g p dn djnh thanh cd lien ket cimg. Men ket ddn hdi da d u g c giai quyet triet de theo nhieu tieu chi dn djnh khdc nhau [1], [2], R. C u n n i n g h a m da giai q u y e t bai todn dn dinh thanh bi nen cd lien ket niia cirng, d dd lien k i t n u a Cling la nhiing lien k i t dan hdi [3]. Gdn day, J. D. A. O c h o a da nghien ciru bai todn dn dinh thanh chiu nen cd lien ket nira cirng cd xet den dnh h u d n g ciia luc cat bdng p h u a n g phdp phdn tir hxiu ban, tuy nhien md hinh lien ket niia ciing vdn Id nhirng lien k i t dan hdi [4].
T r u d n g h g p thanh cd li§n k i t phi tuyen tai bien, bai todn dn djnh t r d nen rat khd khdn ddi vdi p h u a n g phap giai tich truyen t h i n g . T r o n g nghien ciru nay, tdc gia di vao xdy d u n g mdt p h u a n g phdp tinh gan d u n g de giai q u y l t mgt each hieu qua bdi toan dn djnh ciia thanh chiu nen cd lien k i t dj h u d n g (mot t r u d n g hgp rieng ciia lien k i t phi t u y i n ) . C> ddy, tac gia k i l n nghj sir d y n g giai bai todn theo p h u o n g phdp nira ngugc. T r u d c het d u d n g dan hdi ciia thanh d u g c gia djnh dudi dang mdt da thii'c xdp xi bac cao vdi cdc h$ sd cua da thiic la cdc tham sd c h u a bilt.
Viec xdc dinh d u d n g dan hdi ciia thanh t h u c chat Id viec di xdc dinh cdc h e sd ciia da thirc xdp
Tap chi KHOA HOC GIAO THONG VAN TAI So 49 - 12/2015 33
Cong trinh Khoa hoc
xi. Dilu kien de xdc dinh cdc tham sd ndy Id: da thiic xdp xi phdi thda man phuang trinh vi phan CO bdn ciia bai todn dn dinh; ddng thdi nd cung phdi thda man dilu kien bien ddng hgc va tTnh hgc ciia thanh. Md hinh bai todn tdi uu hda dugc sii dung de xac dinh cac tham sd chua biet ndy.
Cac gid thiit ciia bdi todn trong nghien ciru nay nhu sau: Mat cdt ngang khdng ddi dgc theo chilu ddi thanh; thanh cd cac lien ket d hai ddu vd chju nen bdi luc dgc true tap trung ddt tai cac dau thanh. D I gidm thieu sy phirc tap ve mdt todn hgc, trong bai toan nay chua x6t dSn dnh hudng ciia luc cat.
2. LIEN KET DI HUCfiSG
Khdi niem lien ket di hudng trong bdi toan ndy Id lien ket trong dd quan he giiia phan luc lien kit va chuyen vi ciia dilm lien kit thay ddi theo do Idn cung nhu chilu chuyin vj ciia dilm tya Ii8n ket. Mdt sd md hinh todn hgc ciia lien kit dj hudng dugc thi hien trong hinh I, d dd mo hinh lien kit mdt chilu nhu trong hinh la, I b, md hinh lien kit dan hdi dj hudng nhu trong hinh
Ic, md hinh lien ket nhieu Idn tuyen tinh nhu trong hinh Id.
0
[/• .
Hinh 1. Cdc md hinh lien kit di hudng [5]
Quan he todn hgc giifa phan luc hen ket vd chuyin vi dilm lien kit ciia cdc md hinh lien kit dj hudng dugc xdc djnh nhu sau:
Vdi lien ket mdt chilu cd khe hd ban dau Ao, ta cd:
N = f ( 4 - A . ) + | | A - A . | Vdi lien kit ddn hdi dj hudng cd khe hd ban ddu Ao, ta cd:
N = ' ^ l L j i : ( A - A , ) + ' ^ | A - A „ |
(I)
(2)
34 Tap clii KHOA HQC GIAO THONG VAN TAI So 4 9 - 12/2015
Tir cac md hinh todn ciia cdc lien kit dj hudng d tren, ta tdng hgp dugc md hinh toan cho lien ket phi tuyin (nhilu Idn tuyin tinh) nhu sau.
N = N„ + ^ (A,j - A,j., )kj_, + (A - A„„ )k„ (3) trong do: A^jld cac do lech chuan (so vol gd'c tga dd) ciia di^m dau doan thang N-A thiij,
N^i la phan luc lien ket ling vdi chuyen vj diem Ii6n kdt A^j.
S.HAMXAPXi
Ham xdp xi dugc su dung de md td gdn dung dudng dan hdi ciia thanh tai trang thai tdi ban, khi dd thanh da bj cong di. Ham xap xi cd the sii dyng mdt trong hai dang co bdn sau ddy:
chudi lugng gidc hodc da thirc. (3 ddy ham xdp xi da thirc dugc lua chgn do tinh don gian toan hgc, dac biet trong cac tinh todn dao ham, vi phdn, tich phdn. Ci ddy, bac vd sd lugng cdc sd hang ciia da thiic khdng bi han che. Mgt each tdng qudt, da thirc xdp xi c6 dang nhu sau:
v ( z ) = ao-Fa|Z + aj2^-(-...-i-a,2'-i-...-i-a^z" (4) O dd, cdc he sd ao, ai,... a,, la cdc tham sd chua biet, cdn dugc xdc dinh.
4. TOI UtJ HOA XAC DINH CAC THAM SO CUA HAM XAP XI
Ham xap xi trudc het phdi thda mdn phuang trinh vi phdn ca bdn cua thanh tai trang thdi tdi han vdi mgi gid tri ciia z trong mien xac dinh ciia ham sd. Khi khdng xet den dnh hudng cua luc cdt, phuang trinh nay cd dang nhu sau [ 1 ]:
d''v(z) P,^ d \ ( z )
dz"* E J , dz" = 0 (5)
Trong do P,h \k lire tdi iian ciia thanh. O day, sir dung tieu chi nang luong trong bai toan on djnh ta xac djnh dugc P,htheo congthiie (6) [I].
J E J , [ v " ( z ) ] ' d z
P , „ = ^ h (6) J [ v ' ( z ) ] ' d z
0
D I ham xdp xi v(z) thda man phuong trinh vi phan co ban (5), ta sii' dyng phuong phap binh phuong tdi thieu. Vdi mgi gid tri cua z trong miln xdc dinh ciia ham xdp xi, cac tham so a, cua ham xdp xi dam bao sao cho tdng binh phuong ve trdi cua (5) ung vdi cdc gid tri khdc nhau ciia z trong miln xdc dinh cua hdm xdp xi, Z[Vl trai (5)]", phai dat cue tieu, vdi dilu kien rang budc la cdc dilu kien bien dgng hgc va tTnh hgc ciia thanh.
Phdt bilu bdi todn tdi uu hda xdc dinh cac he sd ciia da thirc xdp xi nhu sau:
Hdm muc tieu: f(a,) = Z[Vl trdi (5)]" -> min. Biin sd: a, = {ao, a,,... an}^.
Dilu kien rang budc: dilu kien bien dgng hgc vd dieu kien bien tTnh hgc.
T^p chi K H O A H Q C G I A O T H O N G VAN TAI So 49 - 12/2015 3 5
Cong trinh Khoa hoc
5. UlVG DUNG HAM SOLVER GIAI BAI TOAN
Tir da thuc xap xi da gia dinh ciia dudng dan hdi, ta lap dugc bdng tinh cac dai lugng nhu trong bang 1. Cho cac tham sd a, thay ddi, phuong phap Newton dugc su dung de xdc dinh cdc tham sd a, sao cho ham muc tieu f(ai) dat cue tieu.
z d(z) J,(z) EJ,(z) v(z) v'(z) v"(z) v'"'(2) v""(z) Vltrai(5) [Vetrai(5)r
Ham muc tieu f(a,) = S[V| trai (5)]^
Tham sd bien va cdc dieu kien bien ciia bdi todn dugc thiet lap trong he thdng menu cau lenh cua ndi hdm Solver.
Tren co sd thudt todn da gidi thieu d tren, chuong trinh tinh tren may tinh dugc xdy dung vdi irng dung ciia Ham Solver, Solver la mgt trong nhiing ndi ham cua Microfl Excel, dugc xay dung va dua vao sii' dyng tii' phien ban Microft Excel 97. Vdi Solver, ngudi diing cd thi ung dyng dl giai cdc bai toan sau day thdng qua bang tinh Excel: gidi cdc he phuang trinh tuyin tinh, phi tuyin, cac phuong trinh dai sd bac cao, sieu viet, ham mu,,., tim cac tham sd cua ham giai tich xdp xi ciia tap du lieu thdng ke, quan sat nham phuc vy cho viec tinh toan du bdo; giai cac bai toan quy hoach tdi uu [6].
6. THI DU TINH TOAN VA DANH GIA KET QUA
6.1. Thi du 01 - Tinh luc tdi han cua thanh trdn chiu nen, mgt ddu ngam, mdt ddu lien kit dan hdi (Id xo hai chieu dang hirdng)
Bang 2. Cdc so lieu tinh todn thi du 01
a)
'»*|
So do thanh:Chieu dai thanh / (cm):
Du'dng kinh d (cm):
Vat lieu thanh:
E (daN/cm-):
Do cimg ciia 16 xo, c (daN/cm)
Mot ddu ngam, mot dau lien ket 16 xo dan hoi dang hudng
200 cm 8 Thep 2000000
50 Quy udc don vi: chilu dai - cm; luc - daN Hinh 2. Sa dd tinh thi du 01
- Tinh todn theo phuvngphdp giai tich:
Ggi R la phan luc ciia Id xo, Md men udn tai mat cdt cd tga do z. M = -P(f - v) + R(/ - z)
P - 1 R
Tacdphuongtrinh. E J v " = P ( f - v } - R ( l - z ) . D a t k" =—- => v +k-v = k ' f - - ^ ( | - z )
LJ EJ 36 Tap chi KHOA H O C GIAO T H O N G VAN TAI So 49 - 12/2015
Nghiem ciia phuang trinh vi phdn cd dang: v = f ( l - z ) + C| sinkz + C^coskz Dieu kien bien. Tai z = I, cd v = f;
Tai z = 0, cd v = 0 vd v' = 0.
C| sinkl + C, Goskl =0 0 R
p ' tltC, = 0
C|Sinkl + C, coski
cf P f ( l - - ) + C,=0
fkC,
sink! coskl 1 0
• tgkl = k(l ) c
EJ ,kl = k l [ l - ( k l ) - ]
^ ( P j „ = ^ ^ = 32908,81daN
Tinh todn theo phirffng phdp di nglii (trong bdng tinh, quy uac true z co chiiu tii tren xuong, goc dat tai ddu tren thanh):
Da thitc xdp xi: v(z) = ao+aiz+a2Z"+a3z'+a4Z^+a5z' Dieu kien bien khi chay Ham Solver:
Till z = /, cd do vdng bang khdng, gdc quay bang khdng;
Tai z = 0, cd md men udn bdng khdng, lire cat cd do Idn bang phan luc Id xo.
Ket qua tinh toan dugc thi hien trong bang 3.
Bdng 3. Ket qud tinh loan thi du 01
"$
O.OOE+00 a<
-2,73E-13 a j -4 lOE-11
"!
O.OOE+00 a i 1.37E-05
ao -1.97E-03
Pih 31290.17
(PiOgi 32908.81
Sai so 4.92%
z 0 40 80 120 160 200
d(z) 8 8 8 8 8 8
J , 201.0619 201.0619 201.0619 201.0619 201.0619 201.0619
v(z) -1.97E-03 -1.42E-03 -9.06E-04 -4.55E-04 - I 2 8 E - 0 4 3.08E-14
v'(z) 1 37E-05 1 34E-05 1.23E-05 9.99E-06 6.03 E-06 8.62E-17
v"(z) O.OOE+00 -I.5IE-08
•4.06E-08 -7.67E-08 -1.23E-07 -1.80E-07
v'"(z) -2.46E-10 -5.08E-I0 -7.70E-10 -I.03E-09 -1.29E-09 -1.56E-09
v""(z) -6.S5E-I2 -6.55E-12 -6.55E-12 -6.55E-12 -6.55E-12 -6.55E-I2 Ham muc tieu f(a,} =
VT(5)' 4.29E-23 5.97E-23 9.43E-23 I.57E-22 2.60E-22 4.23E-22 3.39E-21
Tap chi KHOA HOC GIAO THONG VAN TAI So 49 - 12/2015
37
Cong trinh Khoa hoc Nhgn xet:
- So sdnh kit qua luc tdi han tim dugc (P,h) theo phuang phdp d l xudt vdi luc tdi ban theo phuang phdp gidi tich (P,h)gt, sai sd Id 4,92%, nhu vay phuong phap tinh de xuat cho ket qua tuong ddi phii hgp vdi phuang phap gidi tich.
- Mdc dii gia dinh dudng dan hdi ciia thanh Id da thiirc bac 5 nhimg ket qua tinh cho thay dudng dan hdi cua thanh tai trang thai tdi ban trong trudng hgp ndy gdn nhu la dudng cong bdc bdn. Do dd khdng cdn phai gid dinh da thuc xap xi vdi bac qud cao.
6.2. Thi du 02 - Tinh lure tcfi han cua thanh tron chin nen, mot dau ngam, mgt dan lien kit dan hoi dorn hirdng.
Bdng 4. Cdc sd lieu tinh todn thi du 02
| j w
Hinh 3. Sa io tinh thi du 02
So do thanh-
Chieu dai thanh / (cm):
Duong kinh d (cm):
Vat lieu thanh' E (daN/cm^):
Do Cling khi nen cua 16 xo (daN/cm) Do cdng khi keo ciia 16 xo (daN/cm)
Mot dau ngam, mot dau lien ket 16 xo dan hoi dcm hudng (chi chju
nen, khong chju keo) 200 cm
8 Thep 2000000
500 0 Quy uac don vi: chieu dai - cm; luc - daN
Da thuc xap xi: v(z) = ao+a|Z+a2Z"+a-;Z^+a4z''+ajZ^
Bdng 5. Ket qud tinh todn thidu 02 as
0 a4
0 aa -5E-09
aj 0
ai 0.000595
a«
0.07933 P»
25164.17 Polt 24805.02
Sai so i.43%
z 0 40 80 120 160 200
d(z) 8 8 8 8 8 8
J , 201.0619 201.0619 201.0619 201.0619 201.0619 201.0619
v(z) 0.0793 0.0558 0.0343 0.0165 0.0044 0 0000
v'(z) 0.0006 0.0006 0.0005 0.0004 0.0002 0.0000
v"(z) 0 OOE+00 -1.19E-06 -2.38E-06 -3.57E-06 -4.76E-06 -5.95E-06
v"'(z) -3.00E-08 -3.00E-08 -3.00E-08 -3.00E-O8 -3.00E-08 -3.00E-08 Ham m u c t
y'"'(z) 0 0 0 0 0 0 eu fl;a,) =
VT(5)' O.OOE+00 -7.5 IE-11 -1.50E-10 -2.25E-10 -3.00E-I0 -3.75E-10 l.OlE-08 Nhan xet: KSt qua tinh cho thiy, luc toi han va ducmg dan hoi cua thanh t^i trang thai tdi ban gi6ng vai truong hgp thanh co lien kgt mot dSu ngam, mpt dau tu do (lire tdi han (P,h) tinh theo phuong phap dj nghi sai khac 1,43% so vdi lire tdi han (P„t) tinh theo phuong phap Ole 38 Tap chi KHOA H O C GIAO T H O N G V ^ N TAI So 4 9 - 12/2015
khi coi thanh cd mdt ddu Men kit ngam, mdt ddu tu do). NghTa Id thanh bj mdt dn dinh khi bj udn cong vl phia Id xo khdng lam viec (ddu cd lien ket Id xo trd thanh dau tu do). Dieu nay phii hgp vdi thuc tl Id khi thanh bi udn cong ve phia Id xo (Id xo chju nen), so dd lam viec ciia thanh la mdt ddu ngdm, mgt ddu Id xo se dn djnh hon so dd thanh mgt ddu ngdm mgt ddu ty do.
7. KET LUAN
,^ Nghien ciru da xdy dung dugc mgt phuong phdp tinh dn djnh thanh chju nen cd lien ket di hudng dua tren each gidi bai todn niira ngugc vd irng dung ham Solver de tim Idi giai cho bdi toan.
Thuat todn vd chuong trinh tinh da dugc kiem chiing qua thi du tinh toan don gidn va so sdnh vdi phuang phap gidi tich vd phuong phdp Ole vdi sai s6 nhd dudi 5%.
Ket qua nghien cim cd y nghTa quan trgng trong cdng tac tinh todn thiit ke dn djnh cac cdu kien cdng trinh chiu nen, ddng thdi day cung Id tdi lieu tham khao tdt cho cdc nghien ciru ve bdi toan 6n djnh ciia thanh vd he thanh chiu nen.
Ngay nhan bai lan ddu: 25/09/2015 Ngay nh$n bai sua: 27/11/2015 Ngay chap nhan ddng bai: 30/10/2015
Tai Ij^u tham khao
[1], Vu Dinh Lai: Sue bin vat lieu, Nha xuit ban Giao thong Van tai, 2007.
[2]. Timoshenko & Gere: Theory of Elastic Stability, McGraw-Hill, 1961.
[3]. Cunningham, R.: Some Aspects of Semi-rigid Connections in Structural Steel-Work, Structural Engrg., 68(5), pp. 85-92, 1990.
[4]. Ochoa, J. D. A,: Stability and minimum lateral bracing for stepped columns with semi-rigid connections including shear effects: I) Theory, Dyna rev.fac.nac.minas voL79 no. 174 Medellin July/Aug, 2012.
[5]. Luang Xuan Binh, Do Xuan Quy, Nguyen Xuan Luu: Tinh toan kit cau c6 lien ket dj hudng bdng phuang phap phan tu hiru hgn, Tuyin tap cong trinh Hoi nghj Co hoc Toan quoc Ldn thir 8, tdp 2, tr. 57- 68, 2007.
[6]. Tran Tri Dung: Excel-Solver cho kj su, Nha xuat ban Khoa hgc vd Ky thugt, 2005.
Phan bifn: PGS.TS. Tran Due Nhiem, TS. To Giang Lam.
T^pchi KHOA HOC GIAO THONG VAN TAI So 49 - 12/2015 39