Ap dung thuat toan ma tran giai cac bai toan
he thanh bien dang dan hoi bang phan mem MathCad
Applyingmatrix algorithm to solve the problem of elastic deformational system by the MathCiad software
Tran Thi Thuy Van
Ng^y nhan bai: 01/7/2019 Ngay sua bai: 25/02/2020 Ng^y duyet dang: 26/2/2020
Tom tat
Hien nay, mon lioc Ctf hoc ket cau giai quyet cac liai toan he thanh bien dang dan hoi co ban la bai toan xac dinh noi li/c, chuyen vj trong he t i n h djnh va he sieu t i n h ; xac djnh cac thong so on dinh he thanh va xac djnh tan so dao dong rieng cua he de giiip viec xac djnh noi liTc va chuyen vi trong nhiimg he chju tac dung ciia tai trong dong. Tuy nhien, neu chi ap dung cac phUong phap truyen thong va thiic hien t i n h toan thu cong se kho gidi quyet du'dc cac bai toan CO so do hinh hpc va tai trong tac dung phdc tap. Trong bai bao, tac gia gidi thieu each ap dung '
thuat toan ma tran de giai quyet cac bai toan va thirc hien trinh t u t i n h toanbang phan mem lap trinh MathCad, giiip giai quyet du'dc cac bai toan phiirc tap va van hieu dittfc ban chat cua cac phuong phap t i n h toan truyen thong.
Kkhoa: Cahoc ket cau, thuattoan ma tm, phon mem lap trinh MothCAD
Abstract
Nowadays, the subject of Structural mechaniG deals with problems of elastic deformational system such as determination of internal forces, displacement;
determination of stability parameters and natural frequencies that help determine internal forces and displacement ofthe system subjected to dynamic loading.
However, if only the traditional methods and manual calculations are applied, It will be difficult to solve problems with complicatedness in geometry and loading.
In the paper, the author introduces how to apply matnx algorithms to solve problems and perform calculations on the MathCad programming software, this helps to solve various complex problems witti understanding of traditional method.
Key words: Structural mechonics,matrix algorithm, MathCad programming software
TS. Trdn Thj Thay Van Bg mon Sire bin vgt lieu - Ca hgc kel a Khoa Xdy dimg
Email tithvan hau@gmail com DT: 0932238019
1. Dat van de
Nghien ciru giai bai loan xac dinh ndi lue va ehuyen vj, xac dinh thdng so on djnh va bai loan xac dinh t i n so dao ddng neng ciia he thanh biin dang dan hdi la ngi dung ca ban eua mdn hpc Ca hpc kit c l u va On djnh - dgng luc hpe edng trinh, ia ea sd linh loan thili k l kit c i u cac cong trinh ky thuat. Cac bai loan nay cd I h l duac giai quyit bang phuang phap giai tieh va cac phuang phap s l . Phuang phap glai lieh giiip vige lim ra cac an so la cac ham nghiem lien tuc, thoa man phuang trinh lai moi diem ciia vung nghiem dang xet. U'U diim cda phuang phap giai tich la cho lai giai chinh xac va dang tin cay. Tuy nhien, nlu chi ap dyng each tinh toan thii cdng thi nghiem giai tich cd t h i xac dinh duac trong nhtrng truang hgp sa do he kit c l u cung nhu tai trong tac dung len he khdng qua phirc lap. Cdn doi vdi cac bai loan phirc tap thi se gap phai nhiJng khd khan nhai djnh v i mat loan hoc. Vi vay, trong nhiiu truang hgp nguai ta ap dung eae phuang phap s i Cac phuang phap sd the hien nhiing uu diim vugt trgi so vdi cac phuang phap giai tich vi ed the giai dugc cae bai loan he khoi, he l l m , he thanh mdt each dd dang ndu sir dyng phln mlm lap trinh linh lean. Do dd, de cd I h l ap dyng cac phuang phap s l vao viec giai quylt eae bai loan ddi hdi nguai sir dyng vua phai cd kiln thuc nhlt dinh v l mdl phuang phap mai vira phai cd kiln thirc lap trinh d mdl mirc dp tuang ddi chuyen sau, Vi vay, tren ca sa van la cac phuang phap giai tich nhung ap dyng each thilt lap bai loan sir dung thuat loan ma Iran giiip giai quylt cac bai loan phire lap ma khdng gap phai su tra ngai nao va nguai doc vSn nlm dugc ban c h i l ciia bai loan, kiem soai dugc lung bude tinh loan mdt each chai che.
Bai bao d l cap lai su ap dyng eiia thuat toan ma Iran vao cac bai loan neu tren cua ca hoc k i t c l u . dua ra cac md dun lap Irinh tinh loan miu ap dyng vao iLPng bai loan cu the.
2. Ngi dung
2.1. Thu$t toan ma tran trong phan tich tTnh cac bai toan he thanh bien dang dan hoi
2.1 1 Bai loan xac dinh ndi luc va ehuyin vj trong he
Phan iich tTnh cae bai toan he thanh biln dang dan hii la xac dinh ndi luc va chuyin vj trong he dudi su lae dung ciia cac nguyen nhan nhu lai trpng, chuyin vi eudng birc gdi lua va su thay doi nhiel do, vv. Phuang phap giai tieh giai quyet cac bai loan nay mgt each tnet de va eho ham nghiem chinh xac trong mdl khoang nao dd. Cd t h i giai quylt bai loan theo 2 huang: theo phuang phap luc va theo phuang phap chuyin vj. Ca sd l;J' thuyet, trinh l u giai bai loan Iheo 2 phuang phap nay dugc trinh bay cu t h i trong [I], Dua vao each ap dung thuat loan ma Iran thiy rang riit ngin duac tuang doi qua trinh linh loan trong viec xac dinh cae he sd va so hang ty do cua phuang trinh chinh t i c trong 2 phuang phap. Cu the la, trong phuang phap luc truyin thing phai ve cac bilu do md men dan vi (bieu dd do luc X[,=1 gay ra trong he ca ban) va bilu d i md men do lai trong gay ra Irong he ea ban, sau dd ap dyng phep nhan bieu dd de lim duge eae dai luang trong phuang trinh chinh tic. N l u ap dung thuat loan ma tran thi chi viec thilt lap eae ma Iran do cae lue Xt,=1 va tai trpng lac dyng ien he ca ban va diing thao tac trong phan mlm lap trinh tinh loan MathCad cd t h i lim ra cac dai lugng Irong phuang trinh chinh l i e mdl each d l dang Diiu dd duge thue hien luang tu niu giai quyll bai loan theo hudng phuang phap ehuyen VL Sa do khli giai bai loan Iheo hudng phuang phap luc va phuang phap chuyen vi dugc I h i hien nhu irong hinh 1,
SO 37 - 2020
45
KHOA HOC & C O N G N G H E
lU b u czc tkicz SB ^ V30 vi tidi Itenc )mdi hoc cm h t caa kieb ibm-so i-a ben. /
ic Jun SD ds mdl. n c SB
1
TbeE%dcB
1 •.>rrlTimr»r ha
-'•-«-•*—>**= 1
1
aiiiiicQii!^hn:d0minEi£i;3C2cpiiaiihic3CibiB|Li donn § ^ 13 tronE HCB j
1
orajoliaMfedDcMptnHnijmiiftamiflm: j
1
M . M * » ~ — = l - . i ^ « . . - = « « * . A l . l i < i . * i . l - X ,
1
1 s _ j a , ™ » t ™ i U o f c i i a , l » , J i , ™ , m i i i . « M > 1
/ saflm. /
Khai bao cac ttiong so dan vio ve kich thuoc hinh hoc cna he, can faen, Ihoog s6 vat lieo, tai tipag aen tic dung leo he
Roi rac hoa so So Tinh sac dmh so lirong p h i n tu so bac sieu ^ons ciia he
1 ihiet lap he c(T ban cu. a he theo hirong phirong phap chuyen n j
lliiet lap c a c m a tian cua noi luc, do tai tFcog nen va cac
1
chuyen v i Z^ bang don vi gav ra trong H C B
Xac dinh c
I
ic he so Clia phirong trinh on dinb
i
GiaihephuunH trinii onjdinlL thu duoc n l i i e m cua he, tinh Ihdng so tdi han P A
Hinh 1 . Stf do khoi bai toan xac djnh noi lu'c va chuyen vi ap dung thuat toan ma tran
Hinh 2. Stf do khoi phan tich on djnh he thanh bien dang dan hoi ap dung thuat toan ma tran
2.1.2. Bai loan xac dinh cac Ihdng sd dn dinh trong he Bai loan xac dinh Ihdng sd dn dinh trong he thanh biln dang dan hdi ehinh la xac djnh tai trpng tdi han tac dyng len he Iheo lieu chi ve dd dn dinh. Cd I h l giai quylt bai loan theo huang phuang phap luc va phuang phap chuyin vj, d day 6k cd i h l ap dyng d i dang thuat loan ma Iran bai bao trinh bay each giai quyit bai loan theo hudng phuang phap chuyin vi. Trinh tu giai va eae phan lu mau ciia phuang phap chuyin vj dugc trinh bay cy t h i trong [1]. Luu y rang, ngoai cac phln tir m i u dugc thilt lap cho nhtrng thanh chl chiu uin ap dung trong bai loan xac djnh ngi lgc va chuyin vj theo phuang phap chuyin vi phin 2.1.1, thi d i i vai bai loan tim thdng s l on djnh can phai thidt lap eae phln tir m l u cho cae phan tir ehju uin ciing keo-nen,
Sa dd khdi giai bai toan xae djnh Ihdng sd on djnh cua he vai su trg giiip cua phan mem lap trinh tinh loan Malhcad dugc trinh bay tren hinh 2.
2.2. Thuat toan ma tran trong phan iich dong cac bai toan he thanh biin dang dan hdi
Phan tich ddng cua bai toan he thanh biln dang dan hoi la di xac djnh ndi iuc, chuyin vj va nhtrng thdng sd can thiit khae do tai trgng ddng gay ra ddi vai he kit clu. De phan tich tinh loan he k i t elu do tai trgng dpng gay ra thi can xac
^ n h mgt so dae trung ddng lue hgc ciia cdng trinh. Mdt trong nhijng dac Injng ddng luc hpc ea ban va quan trpng nhat ddi vai cdng trinh do la tan so dao dgng rieng. Bai bao trinh bay
4 6 TAP CHi KHOA HOC KIEN TRUC - XAY DITNG
each xac djnh tan sd dao ddng rieng bang thuat toan ma Iran ddi vai he thanh biin dang dan hii.
Tan s l dao ddng neng ciia cdng trinh ed t h i dugc xac dinh theo nhilu phuang phap khac nhau, mdi phuang phap d i u the hien nhirng uu va nhugc diem rieng, phy thude vao sa d i tinh toan va cac gia thilt ap dyng Sdi vai tirng loai he.
Ddi vdi he thanh biln dang dan hdi thi xac djnh tin s l dao dgng rieng theo each ap dung ma Iran dp cung va sir dung Ihuat loan ma tran cho phep d i dang Ihuc hien ban ea.
Trong [1], trinh bay cu the trinh l u tinh loan va sa 3h khdi ap dyng phin mlm lap trinh linh toan MathCad sir dung thuat loan ma Iran xac dinh t i n so dao dpng rieng ciia hf thanh.
3. Vi du tinh toan
Cho he dam sidu tmh chiu tai trong va chuyen vj cuong birc nhu hinh 3. Biit" Md dun dan hdi cua vat lieu £=2.10*
(kN/cm^), mdmen quan linh eua t i l l dien cac th^-^h trong h§
la hing s l l=10^(cm''), gia tri iai trpng tac du, 9 p=6 (kN).
q=10 {mim). Mo=20 (kNm) va chuyen vi cu&ng birc A =0 (mm). Kich Ihuo'c cac thanh I^^S (m), 13=2 (m), 13=2 (m), 1^=4 (m). Yeu d u x a c d j n h ndilucciJahe (Hinh 3)
Trinh i u linh toan ap dyng thuat loan ma trg-, theo huang phuang phap luc su dung phln m l m lap trinh i/?;hCad nhi/
sau;
0RIGIN:=1
T T T
Hinh 3. Vi du ve stf do he dam sieu tTnh Mo
i ,rT ] I
® o ® ®
Hinh 4. Vi du ve sd do rdi rac hoa ket cau va he ctf ban cua he
Buac 1: Khai bao thdng so dau vao ciia bai loan (gia In mddun dan hoi vat heu E, mdmen quan linh tiit dien I, kich thuac cac thanh trong he I,,, gia In lai trong lac dyng P, q, M)
P=6 kN - Tai trpng tap trung tac dyng len he;
q=10 kN/m - Tai Irpng phan bo tac dung len he;
Mo=20 kN.m - Mdmen tgp trung lac dyng len he;
1=10^ em" - l\46men quan linh t i l l didn cac thanh trong he,
E=2.10'' kN/em^ - Md dun dan hoi cila vat ligu cac phln tir trong he
li=5 (m), l2=2 (m), 13=2 (m), U=4 (m) - Kich thudc cac phln lir trong he
Iran, cd ma Iran la "mxn", trong dd n ia bae sieu ITnh, m la sd phln lir Ihanh trong he)
P...,.|,.(|.,..x]
M. + p-(l, + x) + ,-l,.[^i-+l,+l,+xl
-(1,+x) - l - x - ( l , + l , + x ) - ( 1 , + x) - ( l , + l , + l . + x ) - ( 1 , + L + x )
Buae 5: Xae djnh cac he so va so hang tu do ciia phuang Irinh chinh lae
Cac he so va sd hang tu do ciia phuang trinh chinh l i e dugc xac djnh Ihdng qua oac bieu thirc sau:
1 := l.,n j : = l .n k.= l .m
Cac he sd phuang trinh chinh l i e dugc linh theo cdng thire:
\J
.,ypfM.(x), M,(x)
Thu duae k l l qua cac he sd phuang trinh chinh tae nhu
3.051x10"' 1.688x10-" 5.093x10"'"^
1,688x10"* l O U x l O " ' 3.819x10- 5.903x10"' 3 819x10"' 1.736x10-
- khai b^o vec ta chiiu dai phan tir trong he Cdng thu'c xac dmh cae sd hang tu do do lai trpng P gay
(m) - khai bao vec la ehuyin vj eudng birc goi tua trong he
Thu dugc k i t qua la cac s l hang tu do ciia phuang trinh chinh l i e do tai trpng va ehuyin vi euang hire ddng thai gay At =0 dp C - Su thay doi nhiet dd lae dung len cac phln
tir thanh
Buac 2: Rai rac hda sa do tinh. xae djnh s i lugng phln lir m, s l bae sieu ITnh ciia he n
- n=3 - So bac sieu tTnh ciia he {so an ciia he theo phuang phap luc)
- m=4 - So phan tir trong he sau khi Ihue hien Td'i rac hda Bud'C 3: Thiet lap HCB cua he
HCB dugc thilt lap nhu hinh 3.2
Buac 4" Thilt lap cac ma tran ciia ndi luc do iai trpng va cac lue dan vj gay ra trong HCB
+ Thill l^p ma Iran md men uon Mp(x) do tai trgng gay ra trong HCB eho cac phan l u Ihanh (mol thanh dugc v i l l iing vdi 1 hang ciia ma Iran, ca ma Iran Id "mxl", trong dd m la so phan ICr thanh trong he)
+ Thill lap ma Iran mdmen uon dan vi l\11{x) do eae lai trong dan vj gay ra trong HCB (mli mdt tai trgng \=^ gay ra dugc viet vao 1 ept, mdi thanh dugc v i l l vcio 1 hang cua ma
Bude 6: Giai he phuang trinh ehinh tic, thu Cwcyc nghiem ciia he
X : = -S-iA) Nghiem cua he thu du'Q'c la
X = 16.23' 33.34 -0.95
BiFgc 7: N6i!i.fc trong he (m6menu6ntronghe) dirge xac djnh nhu sau:
M „ ( x . k ) - M , ( x ) j + £ ( X , • M, ( x ) j , ) Bugc 8- K§t qua ngi Iyc (monnen) trong timg phan ti>
thanh trong he
Bleu d6 noi luc trong tirng ph§n ttr (hlnh 5)
S O 3 7 - 2020
47
KHOA HOC & CONG NGHE
H a H i ' '
Mid("-i3,3;
Bieu do momen uon phan tCr 1
Phan tif' 3
Hinh 5. Ket qua bieu do noi lu'c ciia vi du tinh toan
Bieu do momen uon phan tiy 2
M.d(«-l2
Ma(xl4
So sanh kit qua tinh loan ap dung lhu|t loan ma Iran blng each sir dung phin mim lap trinh MathCad vai kit qua tinh loan blng phln m i m phan tich kit c l u Sap2000 duac k i t qua hoan loan triing khap.
Ngoai ra cac vi du tinh loan ap dyng thual toan ma Iran d l giai eae bai toan khae da neu trong bai bao dugc trinh bay cy I h l trpng [1].
4. Ket luan
Bai bao trinh bay viec nghien cuu sir dung thuat loan ma tran ap dyng vao cac bai loan he thanh biin dang dan hoi va dua ra quy trinh linh toan ey I h l cho cac bai loan bang phuang phap giai tich truyin thing, Cu t h i la, bai loan xac dinh ngi luc va ehuyin vi blng phuang phap luc, bai loan xae djnh ndi luc va ehuyin v\ bang phuang phap chuyen vj.
bai loan xac djnh Ihdng sd on djnh he Ihanh, bai toan xac djnh t i n sd dao ddng rieng he phdng. Tu quy trinh tinh loan da thill lap, thay rang viec ap dung thual loan ma Iran trong thiet iap cac bai loan he thanh biln dang dan hdi giup ngudi si> dung vira nlm virng dugc ban chit eiia phuang phap giai lich truyin thing, vua giam thiiu dugc cac khd khan v i mat loan hoc trong linh loan.
Ap dyng thuat loan ma Iran de giai blng phuang phap giai lich va sir dung phin mim lap trinh tinh loan MathCad, nhdm nghien eiru da thilt lap eae chuang Irinh con giai cac bai toan he thanh biln dang dan hdi. Tu kit qua tinh loan cd I h l thiy rang vide ap dung thual loan ma Iran da khac phye dugc nhCrng khd khan v i mat loan hgc so vai viee tinh loan Ihii cdng, higu qua d l i vai cac bai loan cd sa d i hinh hge va chiu lai Irgng pht>e lap./.
T a i l i e u t h a m k h a o
/, Tran TJij TItuy Van, Di tai nghien cuu khoa hgc cdp Irudng "Ap dung ihugl loan ma trdn giai cdc bdi loan h^ ihanh bien dgng ddn hdi Iheo phucmg phdp gidi lich. Dgi hgc Kien true Hd n6i, 2019 2. Liu Thg Trinh (CB), Ca hgc kit cdu phdn I, NXB KH&KT. 2009.
3 Liu no Trinh (CB). Co hoc kit cdu phdn 2. NXB KH&KT. 2009.
4. Leu Thg Trinh (CB). Bdi lap Co hgc kit cdu phdn i, NXB KH&KT. 2009.
J. Liu Thg Trinh (CB). Bdi Idp Ca hgc kit cdu phdn 2. NXB KH&KT 2009.
6. LSu Thg Trinh (CB), 6n dinh cong trinh. NXB KH&KT 2009.
7. Phgm Dinh Ba, NguySn Tdi Trung, Dgng Iuc hgc cong irinh, NXB
;, 2005.
48
8. VdNhuCdu. Tinh kit cdu theo phuang phap ma Irgn, NXB Xdy dung. 2004
V. Chu Qudc Thing. Phuangph^ Phdn TuHiiu Hgn, NXB Khoa hgc & kl thugt, 1997
1 ft Phgm Dinh Ba. Bdi Igp dong lifc hoc cang trinh NXB Xdy dung.
2003
12. K. Chopi a, Dynamics of structures: theory and applications lo earthquake engineering 2007 Pearson Education. Inc Upper Saddle River. NJ.
TAP CHi KHOA HOC K I E N T R U C - X A Y DITNG