TINH KET CAU THANH CO MAT CAT THAY DOI BANG PHVCNG PHAP PHAN TIT H O I I HAN
Tom tat:
Abstract:
TS. LUCyNG XUAN BINH ThS. D6 XUAN QUY KS. yu NGOC LINH
Bo mon Sue bin vat lieu, Khoa Cdng trinh, Tru-ang Dai hoc Giao thong Van tai Ket cau thanh co mat cat thay doi dwg^ wng dung kha pho bien trong ky thuat cong trinh nhw: thap cau treo, tru cau, dam cau,... v&iy nghia ve tinh hap ly ve mat chju lye cua ket cau. De tinh toan loai ket cau nay mgt each chinh xac, thong thw&ng phwcfng phap giai tich dwyc sw dung. Tuy nhien, bai toan tr&nen kho khan han va nhieu khi khong giai quyet dwac doi v&i nhwng ket cau phwc tap. De khac phuc diem yeu nay cua phwang phap giai tich, phwang phap so ma tieu bleu la phwang phap phin td'hQ'u han (PTHH) dwac sCfdung nhw mgt cong CIJ hwu fch. Cho den nay, de tinh ket ciu dang thanh co mat cat thay doi thi trong phin l&n tai lieu ve phwang phap (PTHH)[1,2,3] deu khuyen cao chia ket cau thanh cac doan
thanh co mat cat ngang khong thay doi dgc theo doan thanh. Do do, de dam bao dwac do chinh xac cin thiet thi tuy thugc vao twng trw&ng ho'p CIJ the ma so phan tie dwac sw dung de r&i rac hoa kit ciu la khac nhau.
Tai nhwng vung mat cat ngang thanh co toe do thay doi l&n thi phai sie dung nhieu phin tw, ngwac lai thi so Iwang phin tw dwgv sw dung se it han. Dieu nay dan den khoi Iwg'ng tinh toan l&n. De giam khoi Iwgng tinh toan, nang cao do chinh xac l&i giai doi v&i loai ket ciu nay, can thiet phai sw dung cac phan tw thanh co mat cat thay doi di r&i rac hoa ket cau.
Trong pham vi bai bao nay, cac tac gia tap trung nghien cwu xay dyng ma tran do cwng phin tw thanh co mat cat thay doi tuyen tinh, xay dyng thuat toan va chwang trinh tinh tren may tinh de tinh toan ket cau thanh co mat cat thay doi.
Frame structures with axially varying section are widespread utilized in structural engineering such as cable bridges'towers, piers, girders, etc.
due to their rationality in terms of strengths. To analyze the structures, there are two main groups of methods: analytical and numerical meth- ods. In complicated cases, it becomes more difficult even impossible to obtain the solutions by the analytical methods. To make goods these shortcomings, the finite element method is employed as an effective tool.
Up to now, in almost materials about finite element method [1,2], the information about the axially varying section elements is very limited.
This paper deals with the establishing the stiffness matrices of elements with linearly varying sections, algorithm and computer program to analyze the structures. The results of this research permits reducing the volume but enhancing the accuracy of the calculation of structures.
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KHOA HOC-CONG NGHE
I.Datvande
Do npi luc trong k i t d u khac nhau dpc theo true thanh, nen mat d t ngang thanh trong thyc t l thudng dugc thilt k l d u tao mdt each thay doi
nhu trong hinh 1. Phint^c6m„
Parabol
cat thay doi
a) Ong kh6i nha mSy. b) Nhjp cau due hing.
Hinh 1. Cac thi du img dung thanh co mat cit thay doi.
Viec tim Id-i giai giai tich cho cdt ong khdi trong hinh l a la kha thi, tuy nhien se la rat khd khan doi vdi k i t d u d u d i m lien tyc nhutrong hinh 1b. Khi dung phuang phap phan tCp hu-u han d l rd'i rac hda d i m c l u ta se dugc cac p h i n tCp thanh cd mat elt thay doi. Khi chilu dai p h i n tCp du nhd va toe dp thay doi cua mat d t ngang phin tCp khdng qua Id'n, ta ed t h i coi nhu p h i n tCp cd mat d t ngang khdng thay doi d l tinh toan. Day la each XLP ly thdng thudng nhat trong thyc t l tinh toan hien nay. Tuy nhien each tinh nay ddi hdi phai rd'i rac hda k i t e l u vdi s l lugng phin tCr r i t Id'n, va Id'i giai thu dugc se cd sai so nhilu do sy khac biet v l hinh hpc giO-a kit cau thyc va md hinh tinh.
Hien nay cd mdt so p h i n m i m may tinh thuang mai cung da cho phep xet td-i sy thay doi cua mat elt ngang p h i n tCp. Nhung s u tinh toan eua cac chuang trinh may tinh thuang mai nay nhu la nhij-ng hop den bi mat. Cho td'i nay chua cd cdng bo k i t qua nao v l ma tran dp cCrng cua phan tLP cd mat d t thay doi.
Trong pham vi nghien CCPU nay, cac tac gia tap trung nghien CCPU xay dyng ma tran dp cCpng p h i n tLP thanh cd mat cat thay doi tuyin tinh trong cac trudng hgp chju luc ca ban nhu keo (nen), xoan, uon. TLP dd Crng dung cac ma tran dp cCpng phan tu- nay vao xay dyng thuat toan va chuang trinh tinh toan tren may tinh de tinh cac k i t d u thanh cd mat cat thay doi.
2. Xay d^ng ma tr|n 6g cirwg phan tiip thanh c6 m^t cat thay doi tuyen tin h
(3 day ma trgn dp cu-ng cua phan tCp thanh cd
mat cat thay doi tuyin tinh dugc xac djnh theo phuang phap chuyen vj. Hinh 2 bieu dien mpt phan tLP thanh cd mat cat ngang thay doi tuyin tinh, b l rdng mat d t ngang phin tCp khdng doi, chilu cao mat d t ngang phan tCp b i l n doi theo quy luat bac nhat. Cac kich thud-c va he tryc tpa dp phin tLP dugc quy ud-c nhu trong hinh ve.
fy
Hinh 2 Phin tie thanh mat cit thay doi tuyin tinh Theo phuang phap chuyin vj, ta xem nhu p h i n tCp bj ngam chat hai d i u . L i n lugt cho cac lien k i t d i u p h i n tCr nhu-ng chuyin vj dan vj, theo cac phuang lien k i t cdn lai, ta se xac dinh dugc cac thanh phan phan lye tuang Cpng. Theo djnh ly tuang hd, cac thanh p h i n phan lye tuang Cpng dd chinh la cac thanh p h i n cua ma tran dp eCrng cua phin tCp thanh e i n tim.
2.1. Phan tii' thanh co mat cat thay doi tuyen tinh chju keo (nen)
Md hinh p h i n tCp thanh cd mat elt thay doi tuyin tinh chju keo nen dugc t h i hien tren hinh 3.
Bang phuang phap chuyin vj theo md ta d tren ta thu dugc ma tran dp cCrng phin tCp thanh nhu trong cdng th CPC (1).
[k]:
P
y
— 5
x _
., a_
P x
Hinh 3. Phin tir thanh chiu i(6o n§n
trong dd b la be rpng m^t cat ngang thanh, E Id md dun dan hoi cua vat lieu thanh.
2.2. Phan tu' thanh co mat cat thay doi tuyen tinh chju xoan
Md hinh phin tCp thanh trdn cd mat d t thay doi tuyin tinh chiu xoan dugc thi hien tren hinh 4.
Bing phuang phap chuyen vj ta thu dugc ma tran dp cCrng phin tCp thanh nhu trpng cdng thCpc (2).
[K]- K., K
22(2) trong dd:
K22 - K^2 - STIGJ, (^2
32aidf+d,d2+d;)
^ 2 1 - - - ^ 1 1 =
- 3nGd^ dj
Z2a{dl +d^d2+dl)
My
y
X ..
^1 _
a
M ^
„, u
X
Hinh 4. Phin tie thanh tron chiu xoin
vd'i G la md dun dan hoi trugt cua vat lieu.
2.3. Phan ta thanh c6 mat cat thay do! tuyen tinh chju uon phang
Md hinh phan tCp thanh cd mat d t thay doi tuyin tinh chju phang trong mat phang xOy dugc thi hien tren hinh 5. Bang phuang phap chuyin vj theo ta thu dugc ma tran dp cu-ng phin tCr thanh nhu trpng cdng thCrc (3).
[k]=
K, K,
^n ^ 1 3
K,
A 2 2 ^ 2 3 A 2 4 -^31 ^^32 -^33 ^34
^ 4 1 ^ 4 2 -^43 -^44
(3)
Hinh 5. Phin tir thanh chju uon phing
trpng dd: > _ ^ +^2
" {A{h,+h2)-\-Bah,)
'^n - ^21 ~
ah^
^31 ~ ^13 ~
{A{h^+h2) + Bah;)
^41 ~ "^14 ~
^22 -
{A{h^^h2) + Bah^)
a/?2
A(hi+h2) + Bah^
a^h, H—ah.
' B
A{h^ +h2) + Bahj^32 ~ ""23 ~
"^42 ~ '^4-
ah.
{A{h^+h2) + Bah^)
— a}i2 A
^33 =
N 3 ~ ^34 ~
A{h^+h2) + Bah^ B
/ z , - h / Z j
{A{h^+h2) + Bah^)
-aK
A{h^ -I- /?2) -I- Ba\
vai:
A=
^44 -
\2.a^
a^h^ H—a/z, ' B ' A{h^ +h2) + Bah^
b.E.(h2-fi^y
h,
1
B =
/zj - /z, 2/Z2 2/z,
6.a' b.E.h,.h2
2.4. Phan tu" thanh co mat cat thay doi tuyen tinh chju lire ket hap
Ma tran dp cCpng phan tCp khung phang chinh la to hgp tLP ma tran dp cCpng phan tu- thanh chju keo (nen) va phan tCp thanh chju uon phang. Ma tran dp cCrng phan tCr khung khdng gian chinh la to hgp tu- ma tran dp cCpng phan tCp thanh chiu keo (nen), xoan va uon trong hai mat phang quan tinh chinh trung tam cua thanh.
TAP CHi CAU DUUNG VIET NAM
KHOA HOC-CONG NGHE
3. Xay dyng thuat toan va chu-o-ng trinh may tinh de tinh ket cau thanh cd mat cat thay do!
Thuat toan va trinh t y tfnh toan trong bai toan nay cung giong nhu trong phuang phap PTHH thdng thudng. Diem khac biet d day la ma tran dp CLPng phin tCp se dugc SLP dyng theo cac cdng thCpc (1),(2)va(3).
Chuang trinh tfnh toan tren may tfnh dugc xay dyng tren ngdn ngQ- lap trinh Visual Basic 6.0. He thing giao dien cua chuang trinh dugc thilt k l tryc quan, tien dung, than thien vd-i nguai SLP dung.
4. Thi du tinh toan va danh gia kit qua
4.1. Thi diJ 1. Tinh toan thanh co mat cat thay doi tuyen tinh chju keo (nen)
Thanh chju keo dung tam cd mat d t ngang hinh chu- nhat ( b l rdng b khdng thay doi, chilu cao h thay doi bac nhit) nhu trong hinh 6. So lieu tfnh toan nhu sau: E = 2.10'kN/cm\ h1 = 20cm, h, = 10cm, b = 10cm, a = 500cm, P = 50 kN. Yeu e l u tinh chuyin vj dpc true tai mat d t d i u ty do cua thanh.
Tiln hanh tfnh toan theo ba each: theo phuang phap giai tich; theo phuang phap PTHH SCP dyng phan tLP cd mat d t khdng thay doi (phuang phap PTHH thdng thudng); va phuang phap PTHH SCP dung phin tCp cd mat d t ngang thay doi tuyin tfnh (phuang phap PTHH d l nghj). D l so sanh danh gia kit qua, rieng phuang phap PTHH thdng thudng dugc tfnh toan theo nhieu phuang an chia phin tCp (so lugng p h i n tCp thay doi tCp 5 den 50), phuang phap giai tich va phuang phap PTHH d l nghj chi tiln hanh tinh toan vd'i sa d l 1 phin tCp.
Kit qua tfnh toan dugc t h i hien trong bang 1.
Hinh 6. Sa do thanh chju l<eo (nen) Bang 1: So sanh kit qua tinh chuyen vj dau t y do thanh chju keo (nen)
PNwngpMp PPPTHH
PPPTHH
PPli*tk*i
GiA [rf (on) Sai sA (X) Gtl bl (cm)
Gii tn (cm)
SA Dhar> ti>
1
0.0866
0.0866 5 0.0807 6.8602
10 0,0836 3.5132
20 0,0851 1.7819
30 0,0856 1.2049
40 0,0859 0.8586
50 0,086 0 0,7432
Bang 1 cho t h i y khi dung phuang phap PTHH de nghj, chi d n 1 phin tCp thi k i t qua tfnh da xap xi bang k i t qua phuang phap giai tfch (sai so ty ddi ia 0,0507 %); trong khi dd n l u dung phuang phap PTHH thdng thud-ng thi vdi so phin tCp SLP dung d l rdi rac hda k i t d u la 5 thi sai sd la 6,8602%, khi chia td'i 50 p h i n tCp nhung sai sd v i n cdn tdi 0,7432 %. Nhu vay chCpng td phuang phap PTHH d l nghj lam giam khoi lugng tinh toan r i t nhieu so vd'i phuang phap PTHH thdng thud'ng va ddng thd'i lai dat dugc dp chinh xac cao han.
4.2. Thidu 2. Tinh toan ciu dim hop lien tuc co chieu cao mat cat ngang thay doi
c l u d i m hop lien tuc 3 nhjp cd b l rdng b = 12m khdng doi, chilu cao h thay doi theo qui luat parabdl nhu t h i hien trong hinh 7. K i t qua dp vdng va md men uon vd'i tai trpng la mdt lye tap trung dat tai giu-a nhjp duge t h i hien tren hinh 8 va 9. K i t qua khao sat sy thay doi gia trj dp vdng Idn nhit trong d i m theo so lugng p h i n tCr duge SLP dyng (4,8,16,20,24,32... p h i n tCp) d l rdi rac hda k i t elu t h i hien tren hinh 10 va bang 2. Cd t h i thiy khi dung phuang phap PTHH d l nghj, k i t qua bai toan hdi ty nhanh, ngay ea khi so phin tCr dugc SCP dyng ehi la 24 hay 32. D i l u ma khdng t h i dat duge vdi phuang phap PTHH thdng thud-ng.
Parabol bac 2
84.5 m
L
D D
Hinh 7. Mo hinh tinh toan ciu dim lien tuc
m
Hinh 8. Biiu dd dfHong
biSu do : mfemenl [giS tri I6n n h a t : 17092B 8968374
tri ntiO nhat:-300J21 1031626
m£B
Hinh 9. Bieu do mo men uon
Bang 2. Gia trj do vong lo-n nhat theo so phan tu-chia
STT
1 2 3 4 5 6 7 8
S6 ph§n til'
4 8 16 20 24 32 36 40
D p vong lan nhat trong d a m (cm)
0,02 39482 0,0346875 0,0388954 0,0395290 0.0400960 0.0402395 0,0403385 0,0404098
c c
a>
lO c
>
<o- O
0.05 0.04 0.03 0.02 0.01 0
i
y
4 8 16 20 24 32 So phan tir chia
Hinh 10. D6 thi bieu di§n gia tri do vong theo so phan tir chia 5. Ket luan
K i t qua nghien CCPU da dat dugc la xay dyng ma tran dp cCpng cac p h i n tip thanh cd mat d t thay doi tuyin tinh trong cac trud-ng hgp chju lye khac nhau; xay dyng thuat toan va chuang trinh may tfnh tinh k i t d u thanh cd mat d t thay doi cho kha nang giam dang k l khdi lugng tinh toan nhung dong thd'i dat dugc dp chfnh xac cao. Do dd, kiln nghj dua k i t qua nghien CCPU vao thuc te tinh toan thilt k l cdng trinhB
TAI Lieu THAM KHAO
[1] Nguyin Xuan Luu: Phuang phap p h i n tCp hu-u han, Nha x u i t ban giao thdng van tai, 2010.
[2] Chu Quoc T h i n g : Phuang phap p h i n tCp hij-u han, Nha x u i t ban Khoa hpc ky thuat, 2000.
[3] Zienkiewicz O. C: The Finite Element Method, M c G r a w - H i l l , London, 1977.
„ r^n'%I'^i«^ . Tfiftife?^^ n&Ji/ /r/0^^
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