Tfnh thanh thanh mong mat cat ngang hor chju xoan bang phUcrng phap phan tiir bien
Boundary element method for calculation of thin-walled beams with generic open section under torsion
Ngay nhan bai: 17/12/2016 Ngay su^ bai: 20/01/2017 Ngay chap nhan dang: 5/02/2017 TOM TAT
Bii bao trinh b i y cich ap dung phifcfng phdp phSn tif bien giai hki toan thanh thsuih mdng mat c i t ngang hd chiu xoan. He phiJcJng trinh gidi bdi toan xac dinh npi Ii^c vd chuyen v\ bdng phiiang phap p h i n tijf bien dupc thilt ldp trin cd sd nghiem gidi tich cua phtiong trinh vi phan c a ban thanh thdnh mong mat cdt ngang hd chju xodn theo ly thuyet Vlaxov. Til h? phUOng trinh dd xay ddng trinh ti/ tinh vd viet chuong trinh tinh ngi lUc, chuyen vi thanh bdng phan mem Matlab.
Til khoa: Thanh thdnh mong mdt cit ngang hd, philcfng phdp phdn tLf bien
ABSTRACT
This paper presents how to apply boundary element method in solving the problem of thin-walled beams with generic open section under torsion. The system of equations which solve the problem for determining internal forces and displacement using boundary element method was established based on analytical root of fundamental differential equations of thin-walled beams with generic open section under torsion by Vlaxov theory. From this system of equations, the authors established the block diagram and computation program to calculate the internal forces and displacement of thin - walled beams using Matlab software.
Key words: Thin-walled element with generic open section, boundary element method
TS. Vu Thi Bich Q u y i n
Gidng vien, Khoa Xay dving, Trifting DH Kiln tnic Hd Npi
ThS. Ngp Vdn Chung
Cdn bd ky thugt, Cdng ty TNHH Xdy diJng va Ket cdu thfip Nam Cifdng
VQ Thj Bich Quyen Ngo Van Chung
1. Dit van de
Cae phuong phap tinh thanh thinh mflng mat cit ngang hd chju xoin hifin nay duoc xiy di^ng trfin eO sd ly thuylt thanh thinh ming eOa Vlaxov. Trong dfl quan he giii'a ham t i i trgng, not lyc, biln dang va chuyen vj dupc the hifin dudi dang cic phuong Irlnh vl phin tudng minh.
Tuy nhifin nghifim giii tieh tudng minh chl cd t h l xic dinh trong mpt sfl Irudng hpp dem gtin. PhUdng phip sd phd bien nhat hifin nay l i phuong phap phan tif hCfu han eho phep giii bii toin he thanh thinh mflng trong trudng hop bat kJ nhung chl cho phip xic djnh chuyin vi v i npi lye tai eic nut phin tif. Dudi day tie gii xin gidi thieu dudng loi giii bii toin thanh thinh mflng mit cit ngang hdchju xoin bing phuong phip phin td bien [1 ] (Boundary Element Method). Khic vdi phUdng phip phin td hou han, phuong phip phin td bifin dUa ra Idi glil Id hdm chuyin vj, nfli lue doctheotruethanh ma khflng can phii chia thanhthinh nhifiu phin td nhu trong phuong phap phan tdhCfu h^n,
2. Thilt lap he phuong trinh tinh n$l lUc vi chuyin vj thanh thinh mong hdchju xoSn bdng phirong phip phin tdbiin
2.1. PhUtfng trinh vl phin co bin cua thanh thinh mdng mit dt ngang hd
Trong SUc ben vil lifiu, thanh thanh mdng ehju xoin dupe xfit giong nhuthanh chju xoin thuin ttiy vdi gli thilt khi bl xoin mit cit ngang efl thl vfinh tu do, do dfl trfin m^t cdt ngang chf tdn tai duy nhit ung suit tilp vi mflt thanh phan ndi luc l i mfl men xoSn thuin tiiy.
Ly thuylt thanh thinh mdng hd chiu xoin ciia Vlaxflp [3] dupc xiy dyng trfin eo sd cfl ke den sU han chl vinh ty do lim phit sinh dng suit phip trfin m^t cit ngang (dupe gpi l i ifng sudt phip xoin-ufln). Cic mit cit niy dupc goi l i m i l eit xoin klm chl hay xoin-ufln. Nhu viy trong thanh thanh mdng chju xoin klm chl ngoai ifng suat tilp do xoin thuin tuy, trfin mit cdt cfln efl Uhg suit phip do sy vfinh khflng tu do eua mit cit v i ling suit tilp xoin-ufln. Cic thinh phin nfli lue bieu thj dng suit phip va Ung suat tilp xoin-ufln trfin m|t eit ngang l i bi mfl men xoin- udn Bw vi md men xoin-ufln M...
Xfit thanh thanh mflng ehiu xo3n ed ehilu dii I ehju tic dung t i i trpng m(x) bao gdm mfl men xoin va bi md men xoin-ufln nhu hinh 1.
M
* ^ 5 7 - ^
Ifinh 1, So fti tii trofig tac dung trf n ttianh ttiJnh m6ng
Phuong trinh vi phin lifin hfi gfle xoin kilm chl 6(x) cua m i l cit ngang doc theo tme x v i t i i trong m[x) cfl dang:
02.2017 nnnrnoT 39
, _ !oi7
E, = -
1 - \ E J .
VcA: E va G la c i c mo dun dan hdi t h d 1 v i 2 eiJa v i t lieu; k la d i e trUng x o i n ufln eua mat e i t ngang; J . , Jw i i m f l men q u i n tinh x o i n va quat ciia m i t c i t ngang.
Cic mfli quan hfi giifa ndi lyc v i chuyen vi
M.(x)--E,j.e-(x)=-^e-(,);e-(x) = - ! 5 : ^ B.(x) = -E,J.e•(x).-5iiLe•(.);e•(^) =
M ^ ( x ) - G J . e ' ( x ) ' " '
\ l . ( M - M „ ( x ) M , ( x )
trong dd M„(x), M.[x), M,(x) - md men x o i n t h u i n tiiy, x o i n - u o n v i x o i n t o i n p h i n dpc theo true x, B-(xl - bt m d men x o i n - u d n doc theo tryc X.
2.2. Nghifim t d n g q u i t cua phUdng t r i n h vi p h i n Theo t o i n hpe g i i i tfeh phuong trinh (2.1) ed nghifim long q u i t :
0=e(x} + e'(\) (23)
Trong dfl:
6 (X ) - nghifim ciia phuong trinh vi p h i n t h u i n n h i t ; 6 ( \ I - nghifim rifing eua phuong trinh vi p h i n khflng t h u i n n h i t . 2 , 2 . 1 . X i c d j n h n g h i f i m cua p h u o n g t r i n h v l p h i n t h u d n n h i t Nghifim ciia phudng trinh vt p h i n t h u i n n h i t (tUong Ung vdi IrUdng hpp thanh khflng efl t i l trpng t i e dung) v i c i c d^o h i m eua nghifim cfl d^ng:
8(x) = C,+C,\^C,ch(k.\) + C.sh(kx) e'(x) = C, +C,ksli(iv\ ) • C^kch(kx) 6"(x) = C,k-ch{k\)*C,k=sh(kx) G"{x) = C.ksh(kx) + C,k'ch(kx)
Khi thay g i i trj c h u y i n vi v i dao h i m c h u y i n v| tai bifin x=0 vio{2.4), eie h i n g sfl tieh p h i n C , C Ci, C efl the v i i t dudi dang sau:
0 - ( 0 )
(2.4)
e(o) = c,+c,,
9 ' ( 0 ) = L, • k C ,
o-(o)» k'c, e-(o) = k c,
c, =e{o)- c, =9'(0)
e(x) = 9(0)+0'(0)x-(chkx-l)^-(shkx-kx)^M|17)
Tis (2.7) n h i n t h i y n g h i i m cOa p h u o n g trinh vi p h i n thudn n h i t phg thupc c i c t h f l n g sfl tgi bifin x=0 eiJa phdn t d , bao g d m c i c t h d n g sfl bfin hinh hoc (chuyen vj) va t m h hoc (md m e n va bi m d men). Vi m i t co hpc nghifim ciia p h u o n g trinh vi p h i n khdng t h u i n n h i t l i dng x d v l d w y f i vi ciia mfli d i l m trfin thanh do c i c tac d d n g eua c h u y i n vi gdc, dgo h i m chuyen vi gflc, bi m f l men xodn-udn va m f l m w i xodn-uon tai bifin x=0.
2.2.2. Xac d j n h n g h i i m r i f i n g cOa p h i A R i g t r i n h v i p h i n khflng t h u i n n h i t
Nghifim rifing cua p h u o n g trinh v l p h i n khdng t h u i n n h i t Id ting xU v l c h u y i n vi eua thanh tif nguyfin n h i n tdi t r p n g t i e d y n g trfin thanh.
Nghifim rifing cda phuong trinh cfl t h l x i c d j n h theo nghifim cua phU0f>g trinh vi p h i n thudn nhSt do quy ludt t i c d p n g cua m f l m e n vd bl mfl men tai bifin x=0 tdi cac d i l m tren thanh c i i n g t u o n g t u t i c ddng cita mfl men v i b i m f l men dat trfin thanh tdi d i l m efl tpa d p x ldn hon vj trf d i l m d i t lyc. Mfl men p h i n bo l i t i p hpp c i c m d m e n tap trung tai mfli d i l m v i sfi dupe x i c djnh b i n g tfeh phdn trfin doan ed t i c d y n g cua m f l men phin b d Ham b i l u dien m d men p h i n b6 trfin hinh 1 c d dgng
\ I = 0 i f x < a j
M = Jmd^ if a, 5 X < a , M-jmd^-Jmd^ if a^ 5 \
C i c t h i n h p h i n h i m c h u y i n v i d o t i c d f l n g ci^a m f l m e n v i bl mfl - T h i n h p h i n h i m nghifim do t i e d d n g bl m d men B:
x < a , 0 ; ( x ) = 0
. , , r c h k ( x - a , ) - l 1
. \ - a , e,
X) = BJ^i '-L^
- T h i n h p h i n h i m nghifim do l i e d f l n g m f l men M:
x s a , 9 ; ( x ) = 0 ;
„. , , rshTc(x-a,)-k(x-a,)1 x > a , 0,(x =M^= i tl—^^ i i i
- T h i n h p h i n h i m nghifim do l i e d f l n g m f l m e n p h i n b d m:
x < a , e ; ( x ) = 0;
e-(o)
(2.5)
k- k'
Thay (2.S) v i o (2,4) n h i n dupe nghifim ciia phuong trinh vi p h i n t h u i n n h i t phu thuflc v i o c i e thdng sfl tat bifin x=0:
(•)(v) = 6(0) + B^{0)x + ( c h k x - l ) ^ - v ( s h k x . k x ) ^ ( 2 . 6 )
Thay e i c quan hfi gida nfli luc va c h u y i n vj Irong (2J) v i o (2 6) n h i n dupc nghifim cua phuong Irinh vi p h i n t h u i n n h i t
cl'l<(x-a;)-l ( x - a , ) '
k'UJ,. 2GJ
GJ,(){v) '1
„ . , , f [shk(x-^)-k(x-4)]
- [shk(x-4)-k(x-?)]
' kGJ chk(x-a,)-l ( x - a , ) '
k'OJ,. 2GJ,.
chk(x-a,)-l (x-a,)"
2GJ,,
C6 the thay nghiem riSng tS rfc hSm b] gian doan tai cac vi tri cic di^m dit tii trpng. O^ bieu dien tfnh giin doan sCf dung ham don vj Heviside [5] va quy udc diu"+" nhu sau>
, , fO. if x ^ a , , (O, if x < a (x-a) =< ; H ( x - a ) - ^
[ x - a , i f x > a | l , i f x > a Khi dd nghidm n§ng chuyen vi goc ciia phi/ong trinh vi phan di/pc Viet dudi dang mdt bieu thiJc
9 - ( x ) .
B[chk(x-a,).-H(x-a,) ] M[shk(x-a.),-k(x-a,).]
G T
Wil^, chk(x-a.) -H(x-a,) ( x - a . ) /
k'CJ,. 2GJ„
chk(x-a,) -H(x-a,) (x-a,)^' k=GJ„, 2GJ„
Nghidm tdng quit cila phuang trinh vi phin cd dang B (0) , , M . ( 0 ) 8 ( x ) - 6 ( 0 ) - . - e ( 0 ) x - ( c h k x - l ) - g p - ( s h k x - k x ) - j ^
B[chk(x-a,).-H(x-a,)] ^ M[shk(x-a.).-k(x-a.)] ,; „ chk(x-a.)^-H(x-a,) ( x - a . ) /
k=GJ,. " 2GJ,„
chk(x-a,)^-H(x-a,) (x-a,).' 2GJ,.
k=GJ„
2.3. Hf phutmg trinh xic djnh tr^ng thii chuyin vf v i n$1 luc ciia thanh
Tit nghifm tfing quit cOa phuong trinh (2.9) xic djnh cic him md men xoin thuin tdy M™(x), bi md men xoin udn B»(x) v i md men xoin udn M.(x) bing cich liy dao him cija him e(x) theo cic quan h# the hifn trong (2 J). Cic kft qui thu dupc c6 thf vift dudi dang ma trfn nhu sau:
-A, 0 1 --^j
0 0 A ,0 0 fl.
A,]
A, A, A,
GJ„9(0) M„(9)
B.(0) M,(0)
B,(x)]
B,(x) B,(x)
B,(x)J
M,(x) ] B.(x) '
' . N ' . W ! trong dd:
, , shkx - kx A, (x) = clikx -1; A, (x) =
/ \ shkx a,j (\) = kshkx; A, (x) = chkx; A; (x) = - j - B,(x) =
B[chk(x-a,).-H(x-a,) ] M[shk(x-a.) - k ( x - a , ) J
chk(x-a.)_-H(x-a,) (x-a;)_'
k-GJ„c l i k ( x - a , ) - H ( x - 2 0 J „
k=GJ,„ 2GJ„
Bj(x) = Bkshk(x-a|) + M [ c h k ( x - a , ) , - H ( x - a , ) J [rshk(x-a,) , . 1 rshk(x-a,). l ]
B3(x) = -Bkchk(x-a|)_-M shk(x-a4)^
rchk(x-a,) - H ( x - a , ) c h k ( x - a , ) , - H ( x - a , ) ]
-•"[ -t' e J
B,{x) = -B.k.shk(x-a,) - M . c h k ( x - a J ,
shk(x-a,) -shk(x-a,)^ « ic
Viit (2.1 D) dudi dang nJt gon
Y(x)=A{x) X(0) + B'(X) (2.11) Trong dd:
Y(x) - ma trin cpt the hien cic thong so chuyin vi ndi Ipc cOa thanh tai diem cd tpa dd x (vficto eho trang thii eiia thanh);
A(x) - ma Iran vudng nghifim co bin eiia phuong trinh vi phin thuin nhat;
X(0) - ma trin ept eae thdng sd chuyen vj v i ndi lUc tai dilm ed tpa dfl x=0 (victo thflng sd ban diu);
B(x) - ma trin cdt eic thflng so tic dung eiia til trpng tai dilm cfl tpa ddx (vficto t i i trpng);
Khi thay eie tpa dfl bifin x=l vao cic ma trgn Y, A, B hfi (2.11) trd thinh hfi phuong trinh dai sd vdi i n sfl l i nfli lyc va ehuyin vi tai niit bifin x=0 eua Ihanh.
3. Trinh ti/tinh ndi lUc v i chuyin vj thanh thinh mdng hd chju xodn t>dng pht/ong phip phan t d bifin
BUdc 1. Rdi rac hda hfi thanh cic phdn td,
Chia theo bifin h1nh hoc he thinh m eic phdn td duoe lifin kit vfli nhau bdl cic nut Dinh sd niit va mui tfin chl hudng xac dinh dilm diu va cudi moi thanh (h'mh 2)
Kmh 2. Rot r x hoa va danh jJ mit phan ti}
Bude 2 . T h i l t l i p he phuong trinh x i e djnh trang t h i i cua he Hfi phuong trinh trang t h i l dUOc ciia he dUoc lap trfin eo sdghep ndi phuong trinh trang t h i i (2.10) ciia tifng phan t d d i duoe rdi rac hda v i o hfi phuong Irinh ehung eiia e i hfi n h u trong (3.1), t h U t u ghep n d i t h u c hifin theo hudng rdi rgc hda da ehi ra d budc mflt.
V,(x) Y ( x )
A , ( x ) X ( 0 |
| X ( 0 ) *
xjnii
B,(x) B,(x)
B.(xl
BUdc 3. Thifit l i p v i g i i i hfi phuong tnnh dai so xac dinh eae thflng sfl bifin ciia c i c p h i n t d .
Hfi phUdng trinh dai sfl duoe x i y d y n g trfin ca sd thay eie g i i tri toa d p bifin x=0 v i x=l eua mfli p h i n i d v i o hfi phuong trinh (3.1) v i nhan dupc:
Y ( I ) = A { I ) X { 0 ) * B ( I ) = . A ( 1 ) X ( 0 ) - Y ( I ) = - B ( I ) Cic i n sfl n$l lye v i e h u y i n vj eua nut d i u v i eufll mdi phdn t d ndm trong hai ma t r i n X(0) v i Y(l}. Khi ehUa g i n ehe d i l u bifin tinh hpe v i hinh hpe tgi nOt, sfl dn sd Idn hon sd phuong trinh. Do d d e i n g i n c i c difiu kifin bifin t l n h hpc v i hinh hpc tgi c i c niit d l khif bdt dn. C i c difiu kifin bifin ttnh hpc dupc x i c djnh dya trfin c i c phuong Irinh c i n bdng lyc v i mfl men tgi niit. Cic difiu kifin bifin hinh hoc duoc x i c dinh dya v i o difiu kifin c h u y i n vi tai bifin cita c i c c i c p h i n t d .
Sau khi d i khd bdt dn c i n thue hifin vific dl c h u y i n thdng sd i n trong vfie to Yd) tdi vficto X(0} d l v i t h i l t l i p hfi
A * ( l ) X « { 0 . l ) = - B ( l ) (3.2) Cic thil t h u i t t o i n hpc d l t h i l t l i p A • { l ) v a X ' ( 0 . 1 ) dupe trinh
b i y chi t i l t Irong [2]
Giii hfi phuong Irinh (3.2) cho phfip x i c djnh c i c thdng sfl nfli lUc v i ehuyin vi tai bifin ciia timg p h i n lU
BUdc 4 . Xie djnh c h u y i n vi v i nfli lye tai d i l m bat ky cua tifng p h i n t i f l h e o hfi (2.10).
4 . V f d u t i n h
Trfin CO SO ly t h u y l l Irinh b i y trfin da t h i l l l i p chuong trinh tinh ndi lyc va c h u y i n vj eho hfi Ihanh t h i n h mflng chju x o i n b i n g p h i n mem Matlab (2] vfli so dfl khfli chuong trinh tinh t h l hifin trong hinh 3. K i t q u i tfnh t o i n h o i n t o i n triing khdp vdi k i t q u i d n h bdng phuong p h i p g i i i llch.
Trong khudn khfl npi dung bai b i o eie l i e g i i xin phfip trinh b i y n g i n gpn k l l q u i l i n h ndi lUC va chuyen vi eho mflt he thanh t h i n h mflng chju x o i n efl kfch thuPc v i chiu lUc n h u hinh 4. Thanh ed do cOng ehju x o i n GJ-> v i d i e trung x o i n ufln k=1(l/m). K i t q u i tinh cac b i l u do nfli lpc v i c h u y i n vi trong cae hinh S-9.
L
Nhpp cac so li?u ve hinh Iipc, vdt li?u, tai trpng Thi^t l?p ma tr^n cac ham c o ban A(x)Thi^i lap ma U^n vec l o lai trpng B(x)
ThiSl l?p ma ir?n cac thong so bien X(0), Y(l). B(t)
T h i a lOp ma tm X ' ( l ) , A*(O.I)
Tinh do npi l^rc \ a chuj en \ i Y(x) =A(x)»X((l)-^B(x)
^ 1
d
hieu do not lire, chuvcn v Kci ihiicHinhi J So do khoi tliuong irmh Hnh noi luc va thuyen vi thanh thanh mong h i chiu xoSn n-2kNm' M-IOIiNTi. m-ilcNinm M-^OkNm ( ' , '
,
,0 : — - 1lOm
i ,' {
i U \
. . . I .
/ / ' , ' ' •
[Vr\ \ \
lOni
• • ,' 1
Kinh 4 So do he thanh thanh m6n9 (hiu xoin
Kmh S. Bifu (Io g « Kin doc Iheo Iruc thanh
Kinh 6. Bieu do bl n u men loan thuan tuy M d i )
A .A-
Kmh 7, Bifu 66 bi nw men xoin ufin iM Mw(kNm)
Hlnh9 Bi^udDm6menxoJnlJn9M.(x) 5. N h i n x f i t
Nfli d u n g p h u o n g p h i p t l n h nfli lyc v i c h u y i n vj hfi thanh t h i n h m f l n g h d chiu x o i n duoc x i y d u n g trong b i i b i o dua trfin co sd ea hpc v i t o i n hpe vQng c h i c . Trong d d hfi phUOng trinh g i i l dupe t h i l t l i p t d nghifim gidl tich cOa phUong trinh v l p h i n quan hfi c h u y i n vj v i t i i trpng theo l<^ t h u y l t t h a n h t h i n h m d n g eua Vlaxov, Do v i y , k i t q u i tinh bdng phucmg p h i p trfin h o i n t o i n triing khdp vdi k i t q u i tfnh b i n g phuong p h i p gidi tich.Oifiu n i y t h l hifin mflt uu diem vupt trfli cua phucmg phap p h i n t d bifin so vfli e i e phuong p h i p sfl k h i c .
TAlllEUTHMriKHAO
[1] PX BanetjH and R. Butteifield, Boundaiy Element Metfiofi in Engineenng Saence.
McGraw-Hill SocACompanyUmited (UK), 1961,
[2) Ng9 VSn Chung, Tlnh thanh thdnh mdng chjoxodn bing phimgpbopphdn tUblin, LuJn vin tiuc si k j M t Truimg Dal hoc KAi I n k H^ NSi, 2016
l i ] V i , Vbsov, Thln-walkdilastlcBeani, Moscow, 1961,
H]ViTti\Bidt(ti:^,Phuongph^ph&ihibiingidibdltodnMhltbarriibi6idangddnhdi.
Tfp 2 - Tti]r^ t i p H&i nghi Khoa hoc toan quAc Co hoc vSt c3n bien dang l i n thil 12, Oa ning, 2015 {Sl HaiapHH B A , KoHauieKKO C H., OdoduieuHbie tpymcuuo e JodcKix uexauuKU, KHCB.:
Ka)n(oBaayMKa,1974.
02.2017B«CiV{|QT 9 3
