^PhU'O'ng phap sai phan hu-u han tinh
j n S i lyc va chuyen vj d i m co d6 cirng thay doi -JThe finite difference method for calculating
^ t h e internal forces and displacements in the changing stiffness beam
%<NgSy nh|n bai: 19/6/2016
»«iWlflsy siia bail 11/8/2016 iNgJy chSp nhan aSng: 12/8/2016
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l i B T d M T A T
"»*B4i bfe xSy dvllig phiMng phip tinh chuyen vi va nOi lUc cia j l j j d i m c6 di Cling thay doi bang phuang phdp sai phan hOn hjn.
*l»(Vi|c » 5 i cac phaong trinh vi phin dUcJc thay the bang he i . i J ' ' ' " ' " ^ " " ^ * ' ' " " " ^^'' ^ ^ '•*"' ''* " ' ° S tliay ffii tai ra6i B , ! " " ' - '^P ' ' ™ * * " > " ^' ^y <i«nB vio lip trinh bang phfa ,i,Umta HatlBb c6 th« tinh n^i lUc v i chuyin vi cia dim a i do
ictlngthay dfii bat Ic^.
m,jphi!0ngghip sai phin hflu h»n.
•!*ABSTR*!CT
yjjThe paper establishes the algorithm fop calculating the internal
^forces Jhd displacements in the changing stif&ess beam by Mtustog the limte difference method Solving differential
^equations are replaced by difference algebraic equations with ,„j|fmctlons of changing stiffhess at each node. Applying this ililialgorlthm into the programming by using Matlab software can
"Jcalculate the mternal forces and displacements m any changing JstiffiiBss beams.
jJllKey words: internal forces and displacements, changing
•^stiffness beam, finite difference method, lrt1niS.TmjngMfPiSm
• Giing vita, khoa XSy dtlng. Trudng DH Xiy DUng Mi& Tiy l,t/TS.Vil Thj Bich QuySn
.Giing viin, khoa Xiy dung, Trumg DH Kiln tnSc H i Noi
" I T S . TtSn-ntj Thiiy VSn
,(|iGi4ngviSn, khoa Xay dtfng, Tniong DH Kiln tnSc Ha Noi
Trifdng My Pham, Vu Thi Bich Quyen, Tran Thj Thuy VSn
1. Gidfthi^u
_ Trong k i t cau cong trinh thong thudng kfeh thUdc m^t cSt ngang cua dam auoc lua chon la khOng doi tai moi nhip. T i , . nhlSn trong m M so bai t o i n t h i f t ke do y t u c i u v l t h i m my kiln t r i e hoSc t i l t k i i m vSt If u can su dung d i m c6 kfeh thudc m j t c i t ngang thay d i l theo quy uat bat ky.JZo hai phuong phap phd bien duoc i p dung tinh toan n6l UC v i chuyen vl d i m cd dd ciJng thay ddi duoc d l c i p den trong cic t i l lieu la phuimg phSp g l i i tich (tren co sd ly thuylt Sdc b i n v i t |j|u [3 41) S , f , S ' ? * " " " J * " " ' ' ° " ' ' ' " ' ' P "^"S phin m i m dng dung nhu iAP 1. a s m han c h i cda e i hai phuOng p h i p n i y l i chi hlSu q u i khi tfnh t o i n n * luc va chuyen vj d i m cd mat c i t ngang thay ddi theo quy luat liac nhat ve chilu rdng hoic chilu cao. Trong trudng hop d i m cd kich thudc mat c i t ngang thay ddi theo quy l u j t b i t ky se gap phii khd khan ve mat t o i n hgc ( i d dung phuong phap p h i p glil tich) hoac phdc tap khi p h i i chia d i m thanh r i t nhilu p h i n t d cd dd cflng thay d l i (sd dung p h i n m i m dng dung SAP). VIn d l cSn giai q u y l t l i dua thdng sd do cung mat c i t ngang thay ddi bat k> v i o t h u i t t o i n g l i i . Bi khic phuc han Che n i u t r i n tac g i i xay ddng phuong phap glii bai t o i n tinh ndi IUC va chuyen vi cda dam ed dd cdng thay ddi b i t k> b i n g phuong p h i o sai p h i n hOu han. PhUOng phip sal phan hdu han ll,2] la mdt phuong phap so d l g i i i cac phuong trinh vl phan thudng hojc phuoiig trinh vl phan dao ham ning. NOI dung chlnh cda phuong phap nay l i b i l u d i l n dao hdm qua gia tn cda ham tai mdt s l d i l m ndt hdu han »a nhd d d chuyen t u viec gidi cac phuong trinh vl p h i n t h i n h cic phuong trinh dal sd B l giai phuang trinh vi phan thay d j o h i m trong cic phuong trinh bang cac sai phan cda h i m tai cic ndt. Phuang trinh cda chuyin vi hoic ngi luc duoc Viet dudi d j n g sal phin tai m i l ndt, b i l u th| quan h i cda chuyen VI tai mdt ndt va cic ndt l i n can dudi tac dung cda ngoai luc. t/u diem cua phuong phap la cd t h i tim duoc nghldm cda phuong trinh hoac he phUOng trinh vl p h i n cua cdc b i l t o i n cd hoc k f t c i u khdng cd nghldm chinh xic dudi dang tUdng minh.
2. Xay dung phUOng p h i p t i n h ndi life v i chuyin vi ciia d i m cd do cung thay ddi b i n g phtAmg p h i p Sai p h i n hdu h^n '
2.1. H4ph„eng trhri, sal phdn thih ndi hjic vd chuyin vl cdd dim
I ^ r I, < ( ) | . ^
^^')
Hinh 11. So do tii trong vi do vong cua iirr, ' ^ t * u ! ' " ' ^ ' " ^' P*"^" ^ ^° ^ ° ' " ^ " ' ' ^ " ^^ '** ™"9 (hinh 2.1) duoc Pii thi bano rar nhifnnn trinh ui nhan '.n,, R 41-
d^M
7^ = '' (2.1)
{22)
(2.3) d ^ y M
dx'""EJ d'y q dx' EJ
Trong 36 M: momen uon c6 chieu duang niu cang t h d dudi ; q : eudng 36 til trong phan bo. chi4u di/Ong huftng t i / dufti Ign tr§n;
y : 36 vong;
£J: do ciJng chong uon.
Ap dung phuong phap sai phan hu^i han [1,21 thay dao ham trong cac phuang &inh Wng c^c sai phan ciia ham tai cac nut nhu sau:
d'yl , A'y„ _ ( y , ^ , - 2 y „ + y„_,]
dx\ ( A X ) ^ ( A X ) '
d'yl __ A'y^ ^Vn.j-'^y.wi+ftyn-'^yn-i-'
(ix)-
(2.6)cac phuong trtnh vi phan (2.1), (2.2} va (2J) se duoc dUa ve cac phuong trinh dal sd. Tai d i l m chia thiJ n, phuong trinh (2.1). (2.2) va (23) cd dang:
I M , ^ i - 2 M „ + M ^ , = h X j M ,
-1 -*iytv.i +6y„ - ' ' y , ^ , + y , „ j = h ^ -
(2.7) (2.8J
(2.9) 2.2. Phuang pbip sai pbdn hdu bgn gidl bdi tadn tinh ngi n6l lUc vd chuyin vieda ddm tinb dinh c6 dd cdng tiiay dal
2.2.1. H^ phuong trtnh d^i so sai phSn
Tfong dSm tinh dinh n6i lUc xSc dinh tilde phUOng trinh can bSng tinh hoc vi khdng phy thuoc vho 36 cUng cua dSm. Xit mot dim 3an g i i n chju u6n ngang phSng, do Cling uon la mot ham Ej{x)=EJ„.f{x), d i d u dai L vi chju t i i trong bSt Icy (hinh 2,2).
ChiadSm thanh n doan bSng nhau, vfti lufti chia h = - . Thay dao ham trong phuong trinh (2.5J bSng sai phan v6i buftc sal phan d^u h, thu dupc:
(1 = 0,1,2, . . n )
C rxTT- XL
Hinh22 Sodotii Irong dim tfnh dinh Trongdo' y,:d6 vong tai niit t h i i ; ; M,. momen udn tai nut thU i;
EJ, • ^ : do cOng tai niit t h i i /;
Bien ddi phuong trinh (2.10). duoc phuong trinh sai phar i ' y = - r i
Wet phuong trinh sai phan cho ^td dc n«Jt tr§n dSm
(i = 0,l,2„..,n-l.n} :T
fl
=0 -». y ,-2y„+y,=-?!la.
' %.
= 1 -> y„-2y,+y,=-A
'!!'
Hay:[K].{Y} = {B) (sTs) Trong 36: [ K ] : ma tr$n hi so cCia h§ phWOng trinh sai p h i n , ,
{ Y } : v^cto bi^n sfi cCia he phUOng trinh sal phan.
| v ) = (y,}' t {B}: vecto hSng sfi ciia hi phuong trtnh sai phan.
2.2.2. B i l u kien biSn glSi he phuftng trinh sai phan hCftj h ? ( ^ Hi phuong trinh sai phSn (2.12) g6m n+1 phiAmg trinh vfiti]
so. €)4 gidl 3uac h§ phUOng trinh sai phSn c i n phSi bfi sung i l l
bi§n, ^ liin kitngdm: c6 do vong vi gde xoay bSng khfing
y,=o
j y i = o [ y i ' = o L y i . i - y i - i = o lyin^Yi-i Liin kit gdi: c6 it$ vdng vh momen bSng khflngfyi=o Jyi^O Jy,=0 P
(?;•
M = 0 [yi.,-2y, + yi„=0 [y^.^-y^,
Cdc trudng hgp diiu ki^n biin cOa ddm - Ddm liin kit hai ddu gdi tUaVift phuong trtnh sai phan cho cac d i l m niit (2.12) vi kft htfi kien bien (2,17). thu duoc hi phuong trinh dal sfi (n -1) phUOogfl (n-1) Sn so dang ma tr^n [ K ] . { Y } = {8} vftl:
-2 i 0 0,..
0 0 0 0
-2 0 .,0
...,0
-7
' f ) 'ii. y, y, y.-y,.,y„)' (2.19)
' " j'l T " f • ? - ^ 1 '"»'
" - C)dm liin kit ddu ngdm. ddu tudo
m phuang trinh sat phan cho ^ c d j ^ nflt (2.12) va k4t hop dieu iu^n bf&n 0,16), thu duoc he phuong trinh dal so (n+I) phuong trinh va ((iM-IllnsodangmatrSn [ K ] . { Y } = { B } vfti:
' 2 0 0 0 0 0.„ „ . 0 0 0 - 2 1 0 0 0 0 1 - 2 1 0 0 0.,..„
. . O O I . . 0 0
0 0 0 0 „ . l -2 1 0 0 0 0 0
{^^l ={y, y. y, y . - y , -y., y. y„.,}' (222)
.« '^ TTT"T""ir U '"''
" ' * * Gill he phuong trinh, tim dUOc chuyin vj dUng (do vdng) y, t?i mat m^ !c3t ngang cft hoSnh dd x,. Di tinh dO vdng thyc t4 thi can phai'bien dfii
itlieo cflng thflt 2.11 c, dugc;
^'"^'"EL (2.24) iiplm
2.3. Phutmg phdp sal pbdn bdu hgn gidl bdi todn tinh ndi lUc vd Kbayin v / o l o ddm sliu tfnh cd dd cdng thay ddi.
ialpiiaiiiflili 2-3-1- H§ phuong trinh dal sd sai phSn tinh chuyen vi ciia d3m ftim^ta* X^t m$t dam chju uon ngang phSng cd lien kit hai dau ngSm dfi '^m^<tnq ufin la mdt hhm Ej(x) = EJ,.f ( x ) . chl4u dai L va chiu tai trong bH
I * Stem Mnh 2.3), gklffli)
XL XL
Hinh2J. Sudfiiiitrongdlmsieuttnli
•mU^i n^t"',^^^'-^^^ P^*"3 lien k i t ngam ben trai va thay b3ng cac
<i.\lW^^ l i ^ li§n kit Rg va Mo (hinh 3).
C n^ m .
Hinh Z.4. Glii phdng lien ket tha/ bing phin lUc lien ket dugcphuOngtrinh{2,10)vdi: '^ uu nan nnan
M,:momen ufin tai niitthU/va M, = M, + H^.,i,+M„ ( v ^ M „ , : momen ufin tai niit thU i do tai trgng ngo^i gay ra)
Sau khi b i i n dfii thu duoc:
Hay y M - 2 y , + y ^ , - ^ R „ - A | y f ^ Trongdo: y^=—l.y^
V i l t phuong trtnh
= 0 , I. 2,..., n - l , n)
R»=R<,h (2.2SC) sai phan cho tat d dc nut trfin dSm
- • y o - 2 y , + y , - - - R „ -
r M . =
H a y : [ K ] . { Y | . { B | ^-.^^.^
232. Biiu ki&n bi^n glai he phuong trinh sal phan hifu han Tuftng t u each giai dam tinh djnh, d4 giai diroc h§ phirmig trinh sai phan dam si*u tinh c^n phai bo sung d i l u kr^n bten.
Cdc tn/dng hop cda diiu ki^n biin cda bdi todn siiu tinh - Dam liin kit hai ddu ngdm
V i l t phuong trinh sai phan cho dc d i l m niit (2,25b) vi k l l hop d i l u ki&n bi4n (2.16). thu dugc h? phuong trtnh d^i sfi (n+1) phuong trtnh va (n+l)ansfidangmatran [ K ] . { Y } = { B } v 6 i :
2 0 _ u
-2 1 0 0 _
1 -2 1 0
0 1 - 2 1 0
0 0....1 -2 1 0 u -
0 0 .0 J -2 - - f, 0 0 0 3 -H
{ ^ i = {y. 92 9j y.-9.~y^, R„ i ^ , ) ' f B l ^ l ^ ^ Mp2_ Mpj^ Mp, tA^, M^
' Dam lien kit dau goi, ddu ngdm
Viet phuang tnnh sai phan cho cac diern n
?n bien (2 16&2 17), thu dUoc he phuong tr n an 50 dang ma tran [ K ] . ; Y | = |B| v6i
t (2 25b) va ket hop dieu ih dai so n phuong tnnh
i B i H ^ :
^
(2 32)Giai he phuong tnnh tim duoc chuyen vi dung (do vdng) y, tai mat dt ngang co hoanh do x, va phan luc lien ket R,, M. De tinh 66 vdng
va phin luc thuc te thi cin phai bien doi theo cong thuc 2 25c thu duoc
X-. R.,^5
' EJ„ (2 33)
(2.34) va luc cat] tai mat cat dOi< = h i duac xac dmh dua vao ngoai lUc tac dung V vua tim duoc.
M(x,]=M„+R„.x,.^M,(x,l Q(>i,) = R„ + Q„(x,l(2 35]
3. Thiet lap thuat toan giai
Tren co sft phuong phap g i i i bai toan dam co do cung thay i36i da xay dung tai muc 2, su dung phan mem Matlab lap trinh tmh noi luc va huyen vi cua dam So do khoi va t h u i t toan giai duoc trmh bay chi tiet trong [5] theo trinh
Buftc 1 Chia ludi sai phan va khai bao thom BUdc 2: Xac dinh cac dieu kien bien, Buftc 3: Thiet lap ma tran he sd tdng the phin tren toan the dam [ K ]
Budc 4: Ooi VOI dam tinh dinh Xac dinh vecto noi luc
Ooi VOI dam sieu tTnh Xac dmh vecto noi luc i M. |, ' Q ngoai gay ra tai cac diem nut tren dim
Buftc S'Thiet lap vecto hang sd (B]
BU6c6:GiaiphUangtnnh [ K ] | Y ; = ;6J Timvectobienw | Budc 7: 061 VOI dam tinh dinh Xic dmh chuyen vi \y\
ban dau
he phuong ti
4. Vidu tinh toan Bai toan' Tfnh chuyer P = 20kNddau t u d o , cd 3di (h = 12cm) va be rdng thay doi har
b. = b f - + 00sl (cm)
I inl
\L^.
Hmh 41 Sodd tai ttong vakidi thuflt dam ThUc hien viec g i i i bai toan tren bang ba phuong phap [5] sau diy, ket qu^ tinh chuyen vi tai dau t u do the hien trong bang 4.1:
- Phuang phap tich phan true tiep.
- Phuong phap sai phan hUu han (vdi lufti sai phan thay ddDsiJ dung lap trinh Matlab de g,i,
- Phuang phap phan til hilu han ap dung phan mem SAP 2O00 (trong bai toan dang xet be rong mat c^t ngang thay doi theo quy luJt bac nhat n i m trong pham vi cho phep g i i i cua phan mem
Bang 4.1. Gia tn dd vdng tai d i u tUdo theo tiJng phuong phap PhUOng phap
giai tich
Phuang phap SPHH Phuong ph^p PTHH
-3.03
12 -3.03
Cac ket q u i tinh t o i n the hieu trong bang 4,1 cho thay sir cliSnli lech do vdng khi tinh bang cac phuong phap khac nhau la khdng ding ke. Oac biet siJ dung phuang phap sai phan hCu han cho ket qui cWnli xac hon phuong phap phan tU hdu han cua phan mem SAP Tuy nhlSi neu ham b hoac h thay doi theo quy luat bat ky thi sd dung phirong phap sai phan hCTu han la don gian va hieu qua nhat
5. Ketluan
Trong bai bao da xay dung phuong phap g i i i bai t o i n tinh chuyAi VI va ndi luc cda dam cd dd cii'ng thay ddi bat ky bing phuang phip Hi phan hUu han Ket qua tinh toan cd sai sd khdng dang ke so vdi phiiong phap chfnh xac, Vfti su tra giup cua cac phan mem Matlab de danggili bai toan d i m vdi budc sai phan nhd de dat dUoc cac ket qui dH) chinh xac cao.
TAI UEU THAM KHAO
11] Nguyen Tien Cirong (dich sach eua giao su, pho iien si KHKT T, Kara mm xki), PhdWjfk so trong cohockelciu, NXB Kfioa Hoc ii Ky Ihuat, Ha Noi, 1985
121 Nguyen Manh Yen, Phuang phap so trong ca hoc kel cdu, NXB khoa hot va ky ihullNi NOI, 2000
H I A B AneKcaHflpoB, B i| PoranoB, B n flep*jBHH, (onpomuenenue Mintsmm MocKBa iiBbicuiaa lUKonao, 2 DO 3
H i r e HHcapeKKO,Efl ArapeB,AA K m r K ^ B f OomoKin ImKm.Conpomm/
|5] Tri/ong ^t/fttim, Nghien cUu phuang phop iir,
tl, Luan vJn thae !i ^ thuat, Trirong Dai hor kien iiuc I m chuyen vi ddm co do OJUj *»
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