Johnson,...Dd thuc hien phuang phap phan tir huu han, ngu5i ta timing lan lugt theo cac buac sau : a Chia mi^i dugc x^t thanh cdc phan tu rieng biet, tren cac bien cua m6i phan tir d6 du
Trang 1DAI HOC Q u d c GIA H A N O I TRllONG DAI HOC KHOA HOC TV NHI£N
Va Khic Bay
TCVH T O A N T R A N G T H A I D A N DEO CUA MOT SO K&V
CAU e m u QUA TRENfH D A T T A I PHtfC TAP BANG PHUONG P H A P BI^N rafi N G H l t M D A N HOI
C!)uj<n ngan!) : Co hoc vat r^n bI6n dang
Ma id : 1.02.21
Luan ^n Pho tI6'n si Khoa hoc To6n - Ly
Ngirdi hirorng ddn khoa hoc : Gido su Ti^n sT Dao Huy Bich
Trang 2MyCLUC
DAN DEC VA PHiroNG PHAP BE6J T H £ NGHI^M DAN Hbl
$ 1 Cac h6 thitc co ban m6 ta ly thuy^t qua trinh bi€^n dang
dan deo 7
$ 2 Bai toan hitn va phuong phap bi€h th^ nghifim dan hbi
cua thuylEt qua trinh bi6h dang dan deo 13
2.1, B^i toan bi6n cua ly thuyfit qua trinh bifih dang dan deo 13
2.2 Phirang phap bi^n th^ nghifem dan hbi trong ly thuyet
qua trinh bi&i dang d ^ deo M6 hinh tinh toan 14
TRONG M O T S O BAI TOAN PHANG CHIU TAI PHtfc TAP
$ 1 Trang thai dan deo cua 6ng d^y chiu ap luc trong va lire keo
doc true 23
$ 2 Trang thdi dan deo thanh tron chiu xoan va lire keo doc true 37
$ 3 Trang thai dan deo cua ban trdn mat chiu lire phap va bi6n
chiu keo nen d^i xiitig 46
$4 Nh$n xet cac k^t qua trong churofng II 58
TRONG MOT S 6 BAI TOAN V 6
$ 1 V6 tru ttra tir do trfin bi^n, chiu lire mat phan b<5 va thay doi
tayy 59
$ 2 Trang thai dan deo cua Itrdi d ^ cong c6 dang v6 tru 76
$ 3 Nhanxet 103
TRONG BAI TOAN K H 6 N G GIAN
$ 1 Tru tron ngan chiu tai phiJc tap 104
$ 2 Nhanxet 124
KftT LUAN 125
TAI U&V IHAM KHAO 127
PHULUC
Trang 3Nghien cihj trang thdi img su^t, bifi^n dang trong vat tli^ d^n deo ma
ra tri^n vong su dung ddy du kha nang 1km vi6c cua vat li^u D6i tuong nghifin cuu cua ly thuyft deo bao gdm nghidn cun quy luat ung
xu cua vat lieu khi co bi^n dang deo vk c^c phacfng phfip giai cdc bki
todn cua 1^ tliuyft deo xu^t ph^t tilr c^c ytu clu cua iJitrc ti^n Hifin nay
1^ thuyft deo ph^t triln theo 3 hudmg chlnh [ 68] : L^ tliuyft qud trlnli
bi^n dang dkn deo, i^ thuyft chay vh ly thuyft tnrcrt
Trfin ca so c4c k6t luan cua vat 15^ vk nhmig s6 lifiu tlif nghifim,
ngir6i ta nit ra ducrc c^c quy luat cd b ^ v& cilng vdi cdc h^ time co s6 cua ca hoc vat ran bi^n dang, lap thknli he c^c phirong trinh ca ban cua 1^ thuyft deo VSn d^ r^t quan trong ti^p theo 1& dua ra c^c phiTong phdp giai cdc bki t o ^ deo
M6t dac di^m ro net cua ly thuyft deo la tinli ch^t phi tuyfi^n cua c^c phirang trinh ca b4n, do do khi giai c^c b&i to6n nky t l m ^ g gap phai c^c kho khan v^ mat toan hoc, vi vay ciing vai vide tiin kiS^m cdc phuang phap giai tich ( ma thudng gap cac bi^u time todn hoc r^t phiTc tap ), phuang ph^p g ^ dung dong mdt vai tr6 r^t quan trong C4c phiTOng phdp g ^ dung nky xu^t hifin do yfiu c^u cua time ti^n, no phai
phii hop vdi cac ca so khoa hoc cua ly thuyft, vi^a phai don gian d6 dt
sir dung [30,42, 45, 5 2 ]
M6t s6 phuang ph^p gM dung da dagc dua ra va su dung :
- Plmong phdp nghiem dan hbi trong ly tliuyfi^t biCn dang dan deo nlio
Iliusin [ 30 ]: Day Ih m6t phuang phdp g^i dung duac sir dung rfft
hieu qua, phuang phdp nky cung duac su dung cho ly tlmyC^t chay [4]
Ngucri ta da su dung phuang phap why di giai quyCt hhng loat bki lodn 15^ tlmyft deo [ 34, 46 ] Bang phuang phap nhy ta dua vific giai b&i todn bien phi tuyfi^n vS vific giai liSn tifi^p c^c bai toan cua ly tlmyft dkii
hbi cua vat th^ t h u ^ nh^t, dang hudiig vai tii ngoki va luc kh6i phu
them Sir hdi tu cua phuang ph^p nghiem dhn hbi duac cliihig minli
1^1 dau tien do V.Panfiorov [ 39, 40 ] trong khi ti^n hknh nghien cuu
bai toan bi^n dang dkn deo cua ban \k v6 Qiirng minh su h6i tu
trong tnrdng hop tdng qudt duac I.I Vorovich va Ju.P Crasovsky tli^ hien trong c6ng trinli [15] Do tInh chat cua ham deo co Uioa
Trang 4nghiem hOi tu theo chudn cua khdng gian Hilbert vfe nghiem suy rOng cua bai todn bien Khao sdt t6c d6 h6i tu cua phuang phdp nay da duac D.L Bucov va V.A Satsnhev tifi^n hanh trong c6ng trinh [10 ]
va d^ tang i6c d6 h6i tu, cac tic gi^ da dua ra ham mdi r(*jvdri Tie J = 1 - [1 - oKeJp I G\ trong do G* > - , khi do
a„ = 3G*e^[l - r(e„)] G.L Brovco va V.X Lensky da ma rdng phuang phap tren cho vat th^ khOng thu^n nhat c6 tInh de*n nhiet d6
va bite xa [ 7 ], cdc tie gia da thift lap quan he :
toan cua 15' thuyS^t dan hbi cua vat \hi khdng thuan nhat di hudrng
Chung minh su hOi tu cua phuang phap nay d6i vdfi vat tli^ thuan nliat dang hudng dugc D.L Bucov trinh bay trong cOng trlnli [9], trong
do tac gia chl ra rang : ham deo co can tlioa man
0< o){ej<.a)(ej^—e^<\ D6i vai vat tli^ khOng thuan nhat va di
hudng, chutig minh su h6i tu va danli gia t6c d6 h6i tu cua phuang phap tham s6 dan hbi thay ddi dugc XE Umaruki trinh bay trong cac cdng trinh [49,50]
- Riuang phap bifi'n phan va phuang phap sai phan huu han k^t hgp
vai phuang phap gan dung lien ti5p [ 27, 48 ] Do su phat tridn \^ ky tliuat tInh toan trong nhung nam gan day, ngudi \s\ sir dung nhi^u dC'n
phuang phap sai phan hihi han hoac phuang phdp ph^i tir huu han ke^t
Trang 5Nghien cihi phuang phap p h ^ tu huu han va khao sat su h6i tu cua n6 trong tru&ng hgp bai toan tuyS^n tInh va ung dung vao trong cac bai toan deo da dugc rfft nhi^u cac tac gia quan tam : V.G Konihep, J Deklu, J Jftriang, J.Fikx, XE Umanxki , C Johnson, Dd thuc hien phuang phap phan tir huu han, ngu5i ta timing lan lugt theo cac buac sau :
a) Chia mi^i dugc x^t thanh cdc phan tu rieng biet, tren cac bien cua m6i phan tir d6 dugc xac dinh bai Uiu" tu cac di^m - goi la cac niit cua ludi chia
b) Dich chuy^n trong m6i phan tu dugc tuan tlieo m6t quy luat ham
xac dinh, bao dam su tuang thich bifi^n dang va dang cua tliam s6 dich
chuy^n cua niit
c) Nhd nguyen I^ biefn phan Lagrang ma bai toan dugc chuy^n v^ giai he phuang trinh tuyfi'n tinh d6i vai dich chuy^n cua cac niit
d) Giai he va sau khi nhan dugc cac dich chuy^n, dSn de'n tInh ung suat, bi^n dang trong m6i phan tu
D6i vdi phuang phap nay nguofi ta c6 th^ nhan dugc nghiem vai btft
ky d6 chlnh x^c nao, khi ludi chia du tru mat Tuy nhien, thuc te^ khi thuc hien phuang phap nay se nay sinh m6t loat cac kh6 klian, vi nliu
m6t trong cac kho khan d6 la su lien he giua m6t s6 cac tpa d6 cua
cac nut dang t6 ong va m6t kh6i lugng Idn th6ng tin khdng du chlnli xac
Difeu khac nhau ca ban giiia cac 1^ tliuyfi^t deo la he thuc vat 1^ lien
he giua cac thanh phan ung suat va cac thanh phan biC^n dang Ndi chung, quan he ung suat - bie^n dang trong ly tliuy^t qua trinh biffn
dang dan deo la quan hS phidm hkin , nhung v6i viec dua vao gia thiS^t
xac dinh dia phuang va gia thift dbng phang, da nlian dugc quan he ham trong he thiJc giira ihig suat va bi^n dang [ 68] , do do c6 thd ap dung hieu qua 1^ thuyS^t deo nay vao cac bai toan trong thuc t^ ky thuat Trong 1^ thuyfi't chay cung nhu cac ly thuyft deo tiuac day chi chua m6t ham vat lieu ( ham chay ), nhung trong ly thuyft qua trinli
bie^n dang dan deo thi chiia hai hkm vit lidu, m6t ham dac trung cho
tInh chat v^c ta va ham kia dac trung cho tInh chat v6 huang cua vat lieu, vi vay n6 phan anh dung han su lam viec cua vat tli^ bi6i dang deo khi dat tAi phuc tap
Trang 6dan deo, trong cac cong trinh [ 19,64, 66, 68] , tac gia da dua ra cac phuang phap gan dung : Phuang phap bien the nghiem dan hoi, phuang phap nghiem dan hoi utig vai toe do, phuang phap tham so dan hoi thay doi ung vdi tdc do, cac phirang phap bien phan va phuang phap sai phan hiru han ket hgp vai phuang phap g ^ dung lien tiep, cac phuang phap gfln dung lien tiep ket hgp vai phuang phap phln tir hiru han Mot phuang phap rat co hieu qua trong so cac phuang phap
gan diing dugc neu tren la phirang phip bien thenghiim dan hoi trong
ly thuyit qui trinh biSh dang dan deo [ 19, 66] De sir dung phuang
phap gki dung nay, ta chia qua trinh dat tai thanh nhieu giai doan O m6i giai doan tai, ta phai giai lien tiep cac bai toan dan hbi cua vat the thuan nhat, dang hirong vdi luc khoi va luc mat phu them Cac luc them nay phu thu6c khong nhung vao cac gia tri tinh dugc trong lln lap ke truac ma con phu thu6c vao cac gia tri trong cac giai doan trudc giai doan dang xet
Vdi m6i mot phuang phap g ^ dung, de cd th^ chap nhan dugc hay khong la phai chung minh dugc su hoi tu, khao sat toe do hoi tu cung nhu su on dinh cua nd Su hoi tu cua hai phuang phap nghiem dan hoi
va phuang phap tham so dan hoi thay doi cho bai toan bien ly thuyet qua trinh dan deo ihig vdi tdc do da dugc chung minh chat che ve mat
ly thuyet trong cac cong trinh [65, 67, 68] Doi vdi phuang phap bien the nghiem dan hbi cua ly thuyet qua trinh biSn dang dan deo de xac dinh ban than cac dai lugng chuye'n dich, bien dang, utig suat, do tinh phiJc tap cua cac he thitc, cho deh nay chua cd dugc chutig minh ve mat ly thuyet su hoi tu, tdc do hoi tu cua nd
Chfnh vi vay ma trong luan an nay d^ ra muc dich nham giai quyet hai van de :
1) Sir dung phirang phip biih thSnghidm dan hdi cua ly thuySt qui trinh biSh dang dan deo Ai tinh toan trang thai ung suat, bien dang cua m6t sd ket c^u di^n hinh chiu qua trinh dat tai phifc tap (tiJc la cac
thanh p h ^ tai phu thu6c vao mot tham sd t ) Cac bai toan deo nhu vay la Idp cac bai toan con ft dirge nghien cuu
2) Tren ca so k^t qua tinh bang sd cac bai toan tren, khao sat su hdi
tu, tdc dd hdi tu va sir 6n dinh cua phuang phap g ^ dung nay
Trang 7PbaatD&dau:- Neu tinh chat chung ciia cac ly Uiuye^t deo dSn d€\\
ddi hoi phai cd cac phuang phap gan dung phd hgp vdi nd
- Gidi tliieu cac phuang phap gan dung trudc day trong cac 1^ thuye^t deo va su md rdng cua nd
- Neu tinh chat dac W\h ciia ly lliuyft qua trinli biCn
dang dan deo va su can thift cua phuang phap bi^n th^ nghiem dan h(5i trong 1^^ thuyft qua trinh bi^n dang dan deo
- Neu muc dich, phuang thuc giai quyft van d^ cua
luan ^i\
Cbatmg Ih cac he time ca ban ciia ly tliuyS^t qua trinli bi^i dang
dan deo va phuang phap bi6i tli^ nghiem dan hdi
- Dat bai toan bien cua 1^ thuyft qua trliili bi6i dang dan deo khi chiu tai phuc tap
- Neu phuang phap bie^n th^ nghiem dan hdi trong ly tliuyft qua trinh bifi^n dang dan deo
- Neu md hinh tinh toan cua phuang phap bi^n Uid nghiem dan hbi trong phuang phap tinli
Cbmmg II: Khao sat cac tinh chat dac trung cua phuang phdp
bie'n th^ nghiem dan hdi trong Idp cac bai toan phang chiu tai tlieo quy luat bat ky thdng qua viec giai cac bai toan : trang thai dan deo cua dng day chiu ap suat trong va k^o - n^n doc true; trang thai dan deo
cua tru trdn chiu keo - n€x\ doc true va chiu tac dung cua ni6 men
xoan; trang thai dkn deo ciia ban tron cd 16 hdng chiu dan - n^n tren bien va tren mat chiu luc phap tuy y Nhan x^t chung v^ phuang phap bie^n th^ nghiem dan hdi trong Idp cac bai toan nay
Cbumg III: Khao sat trang tliai dan deo cua he dam ludi dang v6
tru va m&nli v6 tru chiu tai trong phan bd tUy y tlieo quy luat bat ky Nhan x6t chung v6 phuang phap bie'n th^ nghiem dan hdi trong Idp cac bai toan nay
CbmmgIV Khao sat phuang phap bie^n th^ nghiem dan hdi trong
bai toan khdng gian : tru tron ngan chiu tai ddi xung true, nlian x^t v^ ke^t qua tinh toan
K^t luaa : Danh gia cac tinh chat dac tnmg cua phuang phap bien
tlid nghiem dan h^i trong ly thuyfi^t qua trinh bifi^n dang dan deo va klia nang ap dung cua nd, Neu phuang hudng cd th^ tie^p tuc nghien cihi
Trang 8Xemina bd mdn Ca hoc khoa Toan - Ca - Tin hoc, tnrdng dai hoc T6ng
hgp Ha ndi; hdi nghi Khoa hoc khoa Toan - Ca - Tin hoc Dai hoc Tdng hgp Ha ndi 11-1994, hdi nghi Ca hoc toan qudc lan thit nam 1992, hdi nghi Ca hoc Vat ran bififn dang toan qudc lan thir tu 1994 Cac ke't qua chlnh cua luan an da dugc cdng bd trong [59, 71, 72, 73 , 74] Luan an dugc hoan thanh tai bd mdn Ca hoc thudc khoa Toan - Ca - Tin hoc, tnrdng Dai hoc Khoa hoc Tu nhien - Dai hoc Qudc gia Ha ndi
Trang 9CAC H$ THtrC CO BAN CUA LY THUYfiT QUA
TRINH JM£N DANG DAN DEO vA PHl/ONG PIlAP
Trong chuang nay trinh bay cac he tliuc ca ban md ta qua trlnli
bie'n dang dan deo , cac he tliitc cua bai toan bien cua ly tliuyft qua
trinh biffn dang dan deo va ndi dung ca ban cua phuang phap bidn lii^
nghiem dan hdi va md hiiili tinh toan cua phuang phap nay
BI£N D ^ G DAN DEO
Khi xay dung cac md hinh cua cac ly Uiuyft deo, ngudi ta dua IrCn
cdc quy luM ca ban cua vat ly, cua nliiet dOng hoc va cac dac tnnig ca
hoc cua vat lieu Ndi chung, mdt ndi dung vd cilng quan trong trong
cac ly Ihuye't deo la xay dung mdi lien he vSt ly giua ung suat va bie'n
dang khi vat th^ chiu tic dung cua cac difeu kien ngoai : chiu tai, nliiet
dd, buc xa, Dudi tic dung cua qua trinh dat tai thi xuat hien ben
trong vat th^ dan deo qua trinh bie^n dang, ung suat Trang thai ung
suat tai mdt tlidi di^m nao do ( klii vat tli^ lani viec d ngoai gidi han
dang hdi ) se phu tliudc vao lich su qua trinh bi6i dang, tuc la phu
thudc vao qua trinh dat tai Cac ly tliuyft deo trudc day da md la duac
chilmg muc nao day hien tuong dd Lf Uiuydt qua trinli biCn dang dan
deo xuat hien la mdt each tie'p can khac d^ \&y dung quy luM ddi xu
deo cua vat tlie\ phan anh dugc qua trinli ca ly xay ra trong vat lieu Ca
sd cua 1^ tliuy^t nay dua tren dinh dS dang hudng vh nguyen ly cham
tre cua Iliusin [30, 68]
He qua cua dinh dfe dang hudng cho lien he giua cac v^c ta ung suat
va bie^ri dang :
a=a^cosd„.p^ (1.1.1)
cos 9„ cos ^„ = 1 ( n = 1 ^5 )
—• - »
Trong dd a v€c ta ihig suat, cr^ la c&ang dd ung suat, | p J
-la re-pe tu nhien cua quy dao bi6i dang , ^„ - -la goe dinh liudng cua
Trang 10vdc ta ung suat theo 'p „ Theo dinli dl dang hudng thi cr^, 0„ chi
phu thudc vao hinh hoc ndi tai cua quy dao bifi^n dang va theo nguyen
1^ cham tr6 thi chung chi phu tliudc vao lich su qua trinli bi^i dang
cua doan hiru han quy dao bi^n dang trudc đ, doan nay dugc goi la
\A cham tr§ // Do vay â, 6„ Ih phi^m ham cua đ cong, đ xoan x^
cua quy dao biên dang, cua đ dai cung s cua quy dao bi6i dang va
cdc ham vd hudng dac trung cho su thay đi cac difeu kien vM 1^ cua
qua trinh Cac dai lugng nay bat biên vdi ph^p quay va ph^p chi^u, tuc
Viec xac dinh cac tinh chat, cau tnic giai tIch ciia cac phi^m ham va
cac gia tri cua cac dai lugng dac trung trong cac he time ( 1.1.1 )
-(1.1.2 ) gap rat nhiSu khd khan, đi hdi phai cd nhung nghien cuu
tiép theọ Tuy nhien do yeu cau cua thuc tien ky thuat nen can cd ly
thuy^l vira md ta dugc cac qua trinh bi^n dang phuc tap vilra d^ sir
dung,tuc la can cd cac nguyen ly bd sung nliam dan gian hda md hinh
va Ihiét lap mdt sd dang I^ thuyet deo phii hgp vdi thuc nghiCm D^
lam di^u nay, can cit vao nhiSu kft qua tliuc nghiem ngudi ta dua ra
gia thift xac dinh dia phuang [ 68] Thuyft nay khang dinh rang :
Tifc đ thay đi cua cic goc chi phnang cua cic vec ta ung suit
trong rípe tu nhiin cua quy dao bi6h dang —j^ vi tdc đ thay đi
circmg đ ihig suSit ~^ li him ciia cicgii tri tiic th&i cua Ộâ , đ
as cong vi đ dii quy dao bidh dang:
2 - = / „ ( ^ j t , C 7 „ , / ^ , 5 ) ( 1 1 , 3 )
ds
da
Trang 11vdi difeu kien ban dau (J„ = c^^ , 9^ = 0„^ klii ^ = ^o
Qui ^ r&ng cac phuang trinh trong (1.1.3) khdng chua cac phififm ham
ma la cac ham /« va q/ , do vay <T^, 0^ se la nghiem cua he
phuang trinh vi phan thudng phi tuyên (1.1.3) Cac ham / „ va \p
hoan toan dac trung cho tinh chat cua vat lieu trong qua trinli dat tii
phuc tap, nd dugc xay dung tren ca sd tliuc nghiem va phai cd cac tinh
chat d^ dam bao su tbn tai va duy nliat nghiem cua he (1.1,3) Theo
cac sd lieu thuc nghiem ngudi ta dua vao gia thift đng phang cua
vdc ta ling suit, vec tagia sdiing suit, vec ta tdc đ biih dang Vdi gia
thift nay \€c ta ung suat nam trong mat phang niM ti^p ciia quy dao
bi^n dang, nen tilr ( I l l ) ta cd :
a= a„(cos ^p^^i + cos Ộp^)
trong đ ham / , theo gia tliift xac dinli dia phuang, klidng phu thudc
hi^n vao đ cong x ^^ 1^ '^^i cua 6 , O"^ , 5 CBng tlieo cac sd lieu
tliuc nghiem [ 68] g\k thift rang ham ^ trong (1.1.3) klidng phu
Trang 12\ho d6 cong x cua dd cong quy dao bi6^n dang, sir anli hirong cua d6 cong v^o no diroc th^ hien gian ti^p qua goc 6, tuc 1^ :
Nhir vfiiy hkn / v& hkii ^ klidng phu thu6c v&o dang cua quy dao
bie'n dang, chiing \h cdc h ^ dac tnmg ciia cua vflt lifiu
Trang 13he thiJc (1.1.9) dfin vS dinh luat Hooke : ^ ,j = 2Gd
+ (^a trinh dat tAi dan gian : Khi dd cd 0 = 0
Trang 14thay vao (1.1.15) ta dSn vS dugc (1.1.14)
+ Qu& trinh bie'n dang vdi dd cong trung binli: trong qua trinh nay do
gia triG la nlid, nen :
+ Qak trinh cat tai: xky ra chae chan klii 0 = 7t , klii dd :
Lim-^^ ; L / m | ^ = - 3 G tilr (1.1.9) dan v^ : ^, ^2Gi,,
^ * sin p (J <'-* * -' "
Nhu vay lien he ung suat, bieh dang (M.9) - (1.1.12) cd Qi^ md ta cho moi qua trinh bieh dang phitc tap ca dat tai va cat tii, cac he thuc cua 1^ thuyft biefn dang dan deo nhd, cdc he thu*c cua Prandtl - Reuss
va cua Prager cd th^ xem nhu tnrdng hgp rieng cua ly thuyet nay [68]
Trang 150 3 BAI TOAN BI£N VA PHirONG P H A P BI£N n i i NG1I1$M
DAN HbicuALTf TSUYh Qvk TRINH BI£N D^NG DAN D E O
2.1 B^i toan bi^n cua 1^ thuyet qua trinh bi^n dang d^n d e o
Cho vat th^ chi^m mi^n Q cd mat bien S = Su ^ Sa { Su r\ Sa
= 0 ) , chiu tac ddng cua tai trong bat ky : luc klidi A^/(x,t) , x e Q ,
luc mat I>,(xJ)litnS^ va chuyfo vi cho trudc (p, Iren 5^ CSd\ x&c dinh chuy^n vi ti^ixj) , ten-xa bi^n dang ^y(-^,0 , ten-xa ung suat CTfjixj) (trong dd: u eC'{Q)r^O{Q), £,^,a,^ eC'{Q)r^C\Q) vdi Q = Q^S ) trong mi^n Q va vdi moi i e [ 0, T ] tlida man he
phuang trinli sau day :
+ Phuang trinh can bang :
+ Phuang trinh xac dinli:
3 sin 0 " COS 0 sin 0
Trang 16y/ ^ <I>\S)QOS 0^-(3G -(p\s))l 1 - cos 0Y (1.2.5)
{ a > \,fi > 1,0 < 0 < ;r)
+ DiSu kien bien : Cyrij = E^ , ^^^^
u, = (p, , xeS^ (1.2.6)
van dfe chung minh su tdn tai va duy nhat nghiem suy rdng bai toan
bien (1.2.1.1) - (1.2.6) da dugc ti^n hanh trong cdng trinli [ 56, 68] D^
giai bai toan bien (1.2.1) - (1.2.6) ngudi ta da dua ra nhi^u phuang
phap gan diing ( nhu trong phan md dau da gidi tliieu ) va mdt trong
cac phuang phap gan dung rat cd hieu qua la phuang phap bi6i lli^
nghiem dan hdi da dugc dua ra trong [ 17, 64, 66, 68]
2.2 Phuang phap bi^n th£^ nghiem dan hdi trong 1^ thuyet
q\i& trinh bl^n dang d^n deo Mo hinh tinh to^n
Day la mdt phuang phap gan diing dugc phat tri^n tua nhu phuang
phap nghiem dan hdi trong ly tliuye't bie'n dang dan deo nhd Kliao
sat cac van dS vS su hdi tu, tdc dd hdi tu, su dn dinh cua phuang phap
nay cho den nay chua cd dugc mdt chitng niinli v^ mat ly tliuyft O
day se neu ndi dung cua phuang phap gan diing nay va cac chuang
sau se khao sat bang sd cac tinli chat cua nd qua viec giai cac lap bai
toan khac nliau
1 Npi dung phuang ph^p :
D^ sit dung phuang phap g^i diing nay ta viet he thuc lien he bi^n
dang va ung suat (1.2.3) - (1.2.5) dudi dang:
dS,j=^Ade,j+{P-A)^^S,^ (,.2.7)
Trang 17^c^y =(/l<Jy<5«+2GcJ,tJ;,+//y«)j£« (1-2.14)
Trang 18tu = 1 - — ^ , CO, = I- - 5 ^ - y ^ (1.2.15)
3.C/ ,S 3 U
Phuang phap bie^n th^ nghiem dan hdi trong ly tliuye't qua trinli bie'n
dang dan deo dugc tifi^n hanli nliu sau:
Oiia qua trinh dat tai tiT thdi di^ni ban dau d^n tlidi di^ni dang x^t
[0,t] thanli n giai doan : t^= mr , /M = 0,1,2,3, ;2 Rdi rac hda
cac ham clEln tim tlieo tham sd / :
trong do: Auj"^ = u,{x ,t„) - u,{x ,t„_^)
Kill do tilr (1.2.12) tacd tli^vi^t
Trang 19n-l
o^f = Aei"^Sy+2Ge^^ + X A.f, + A„^, (i.2,21)
m = \
Tliuat giai cua phumig phap bie^ri {i\6 nghiem dan hdi nliu sau : BiCt
nghiem cua bai toan d cudi giai doan Ihu n-1 , nliu vay tlico (1.2.20) cdc dai lugng : Ai %,A2 £);,.• A„_] 61; , hoac Uico (1.2.17) cac dai
lugng: A\eij,/^2^ijy-^n-\ ^ij ^^ ^^^^ ^^^ ^^^*' ^^ ^^^ ^'^^^ nghiem
cua bai toan d giai doan tliii* n : tai giai doan nhy se giai gan diing lien tifp , nghiem gan diing tliu k cua bai toan bien (1.2.1) - (1.2.6) phai thda man cac phuang trinli sau :
Trang 20(".*) _
— ( — + —=! )
m=\
Vol diSu kien bidn :
[^e*^5, + 2G^|;'*)]n, = s r - [ ^ f A„5, + A<„*-%1.;,, txfin ^,
« ; " • * > = ^ ( " ) tren 5„ (1.2.25) trong do :
Trang 21klidi va luc mat khac di
Trong nhiSu trudng hgp, tliay cho (1.2.24) ta su dung cdng tliuc
tuang duang theo dang cdng thiic (1.2.18), tiJc la d lan lap tliu k giai
Trang 22^V'e, = HRiu'' + ^i;/*-"][<'*-'^ - <M (1.2.32)
2 M 6 hinh tinh todn :
Vdi nCi dung cua phucmg p h i p big^n th^ nghifim dan hbi da dugc nCu Lrfin , klii dd m6 hinli tinli todn chung cho moi b^i lo^n d^n deo klii dp dung phirang phdp nay se dugc m6 t^ trong luge d6 dudi day ;
Trang 243 MOt s6 diem can ch(i ^ khi tinh to^n:
A) Do lay difeu kien cr„ ^ CT, ta ap dung quy luat deo d^ tfnh toan,
khi O'u^^s tfl ^P dung quy luM dan hdi, nen tliay rang :
N^u t^^ la gia txi cua tham sd tai d^ tinh toan d cudi giai doan dan hbl ma tai day (7^ sai khac <T^ Vhk Idn thi trong klioAng ti]r t^^ deh ''crf+i dang le ta ap dung quy luat dan hdi nliung tilr lan lap thit 2 trd di ta da ap dung quy luat deo Do vay trong each chia n
giai doan d^ tinh todn, ta can chii ^ d^n didm nay vl nd anli hudng dfifn tdc dd hdi tu va sai sd kha nhifeu
B) D^ tinli toan, ngoai van d^ dua vfe cac dang klidng tlii'r nguyen cdn
can chii ^ de^n ky thuat \\t I^ sd Qiang han d^ tinli dai lugng x trong tlch a.x ( hihi han ), nhung gia tri cua dai lugng a la qua
Idn, vugt qua kha nang bi^u di^n dau phay ddng cua may, klii dd
c&i thay a bdi a = va khi dd ta se tinh x - (^^ v tliay
^ ^ c/;(40) ^ ) ^ ^-80 ^ 1 , do vay se klidng cdn tid
ngai trong van d^ su I;^ sd nira
C) CSn lap ra cac tep gia tri sd lieu cd san d^ tang tdc dd tinli toan
cac tep sd lieu nay cd \hi sir dung cho nhi^u bai toan trong tihig
Idp cac bai toan, vf du ta cd tli^ tinli tap cac gia tri nghiem cua cac ham Betxen : ^ i ( x ) , J o W v trong klioang [ 0,1 ] d^ giiip cho viec tinh toan ddi vdi cac bai toan cd nghiem chua cac ham tien
Trang 25CHlTONG n
Tfrf NGHI$M DAN H O I TRONG MOT SO BAI TOAN
PHANG cinu xii PHirc TAP
• •
Trong chuang nay, sijr dung phuang phap bie'n tli^ nghiem dan hdi cua
If th\iy€t qua trinh bie'n dang d^ giai mdt sd cac bai toan phang Qua cac
ke^t qua nhfln dugc bang sd, dua ra cac nhan x^t ve cac tinh chat hdi tu , tdc dd hdi tu va su dn dinh cua phuang phap gan diing nay
TRONG vA LpC K £ O DOC TR^TC
xet dng day, dai, ban kinli trong bang a, ban kinli ngoai bang b, chiu ap luc d^u d trong P(t) va luc k^o doc true Q(t) 6ng Ihiii bang
vat lieu tai ben, khdng nen dugc ( y= 0,5 ), klidng luc klidi
Ta xet bai toan trong he toa do tru Do di^u kien ddi xinig va dng dai vd han nen cac bie^n dang va ung suat la ham cua r va t :
C^c tlianh phan ihig suat, bie^n dang phai thda man cac phuang trinh
va cac di^u kien bien :
dr ' "^ r
Trang 26<^rr =
-Pit) {
0
khi khi
vdi ^\S) = E' khi 5>5o ,<f)'{s)-E khi 9<5o
D^ sir dung phucmg phdp biC^n tli^ nghiem dfin h'di, chia qud trinh dat tai tir th6i di^m ban d^u de'n thdi didm dang xet tli^nh n giai doan, klii dd
ta du0c :
Trang 27n-l
Sl"^ = IGel"^ + X A„^, = 2G.<"' + ^ A„e, +A„e,
vdi A«^^ = i(/?£-n<OA^r =
Thuat giai d day la blft nghiem d cudi giai doan /?-/ ( tuc la da bift
dugc cac ^m^ij ) ta xac dinh nghiem cua bai toan d giai doan lliu n Trong giai doan nay ta giai g"an diing lien tie^p Nghiem g^i diing thu* k
tlida man (1.2.22)- (1.2.25) , tuc la can thda man :
Trang 28A.c» + A'*-"<;„ = B'"-'-'> 'm'-W ' " « "^^It?
TirdiSu kien e^^+e^g +e^ = 0 nfin tir (II 1.2) ta diroc
Trang 29^C".*) _ ^in.k) _ ^(n.k) _ 2Ge^"'^^ - /?^"'*"'^ - ^(n.*-i:
Dem thay vao phuang trinh (II 1.1) ta dugc
Trang 31Thay cac gia tri vao (II.1.7) , (II.1.8) va (II.1.9) ta tinh dugc
||(e*^ -e")^(^^*^ -^i^-'O'+(e*^ -e")^
va gdc ti^p can giira v^c ta ung suat va tie^p tuye^n quy dao bi6i dang;
cos ^^"
^in.k) ^^,n,k)
Tai miSn vM th^ cr^ < cr^ ( gidi han dhn hdi), thi (o^ = co^^0 , h^
phuang tihih giai bai toan tu ddng trd thanli he phuang tilnli cua bai toan dan hdi
Qwy trinh giAi bai toan theo phuang phap tie^n budc : nhung k^t quA
tfch luy d tat ca cac giai doan trudc la ca sd de' tinli cac dai lugng d giai doan sau
Trong mdi giai doan thiJ n , viec tinh toan theo phuang phap lap : d
gan dung thi} k can tinh cac dai lugng R'"'^-'\B^"^-'^ qua cac sd lieu
da cd cua (n-l) giai doan trudc va cac gia tri g^i diing tliu k-1 theo cac cdng thuc neu tren 6 mdi budc c^ri ki^ni tra gia trj cua
cos 0^"'^^ , n^u cos 0^"'^^ ^ 0 - ung vdi qua trinh dat tAi, nC'u cos 0^"'^^ < 0 - ihig vdi qua trinh cat tAi
Trang 320.52696
1.0914 0.8562
: 24
-9.20 3.69 5.49 0.71559
0.54973
1.1534 0.8384
; " : - Q n l J p 3 ' '$Bd'sfi2-3 ,
p = 0.50651 -6,72 3.12 3.60 0.55578 0.50336 1.0371 0.9218
p = 0.50971 -7.66 3.31 4.35 0.64067 0.53511 1.1009 0.9303
p = 0.51176 -9.48 3.77 5.72 0.72401 0.55750 1.1615 0.9119
q = 1.2E-02 1.8E-02 6.3E-03 4.3E-02 3.4E-02 2.0E-02 6.5E-02
q = 3.7E-02 3.0E-02 4.2E-02 1.6E-02 1.5E-02 8.5E-03 7.9E-02
q = 3.0E-02 2.1E-02 3.7E-02 1.2E-02 1.4E-02 7.1E-03 8.0E-O2
^ thuyft do cong tru
= a ) 'Liai?j^7 :I-uiijpT'^
0.98433 -6.93 3.23 3.70 0.54497 0.49263 1.0263 0.9693
1.0253 -8.58 3.70 4.88 0.62523 0.52201 1.0865 0.9848
1.1242 -10.65 4.18 6.48 0.73235 0.56078 1.1678 0.9801
-6.96 3.26 3.70 0.54293 0.49627 1.0269 0.9764
-8.77 3.79 4.98 0.62357 0.52273 1.0858 0.9880
-10.93 4.28 6.66 0.73277 0.56044 1.1682 0.9842
ng binh
; s « 8 5 7 4 "•
4.4E-03 8.8E-03 5.4E-04 3.7E-03 7.3E-03 5.7E-02 7.3E-02
2.2E-02 2.3E-02 2.1E-02 2.7E-02 1.4E-02 5.9E-04 3.2E-03
2.5E-02 2.3E-02 2.6E-02 1.2E-03 6.1E-04 3,6E-04 4.0E-04
Trang 33Hinh II 1.1 Bi^u db cua 3 each dat tai ngoai
Hinh II 1.2 Hinh anh dan - deo phan bd theo dd day d hai giai doan
Trang 34Dang 11.1.2
Kft qua tinh tai cac di^in r = a , vdi sd giai doan chia klidc nhau
a) TInh todn cho dng day dugc ap dung vdi he tiiuc qud
dang cd dd cong trung binh
0
0„
e,.10' e,.10' e.lO'
q,
o
0„
e.lO' e,.10*
ISn Ifip 3
12 giai doan
0.7082 0.5447 1.1206 -5.0189 3.0446 1.9743 0.7155 0.7915 1.2590
-6.1431
2.6809 3.4622
16 giai doan 0.7086 0.5454 1.1210 -5.0165 3.0430 1.9735 0.7155 0.7925 1.2595 -6.1354 2.6792 3.4562
1 ^ Idp
0.7081 0.5445 1.1204 -5.0191 3.0448 1.9743 0.7146 0.7898 1.2577 -6.1513 2.6871 3.4642
0.7085 0.5453 1.1209 -5.0166 3.0431 1.9735 0.7107 0.7862 1.2539 -6.1261 2.6787 3.4473
trinh biCii
sai s6
5 - ^
2.5E-04 3.4E-04 1.6E-04 4.4E-05 6.0E-05 t.8E-05 1.2E-03 :.lE-03 l.OE-03 1.3E-03 2.3E-03 5.9E-04
1.8E-04 2.3E-04 i.in-04 3.1E-05 4.5E-05 1.2E-05 6.RE-03 8.0E-03 4.5E-03 1.5E-03 1.8E-04 2-6E-03
Trang 35Tdi ngoM Dai luong ISn Iflp 3 1^1 lap 4
e,.10' c,.10' e.lO'
30 giai doan 0.7079 0.5452 1.1206 -5.0144 3.0412 1.9731 0.7104 0.7877 1.2546 -6.1154 2.6719 3.4434
0.7078 0.5452 1.1205 -5.0144 3.0413 1.9731 0.7110 0.7885 1.2553 -6.1218 2.6748 3.4470
sai sd
9.5E-05 l.lE-04 5.8E-05 1.7E-05 2.4E-05 5.7E-06 8.2E-04 l.lE-03 5.7E-04 l.lE-03 l.lE-03 l.OE-03
b) TInh toan dugc ap dung vdi hO Ltiuc qua trnih bi^n dang phuc tap,
Tai ngoM Dai luong
e,.10' e,.10*
C 10'
lan I6p 3
12 giai doan 0.7052 0.5441 1.1185 -5.0220 3.0478 1.9742 0.7069 0.7917 1.2552 -6.1584 2.6971 3.4613
\h\ lap
0.7050 0.5445 1.1183 -5.0222 3.0480 1.9742 0.7050 0.7902 1.2540 -6.1662 2.7030 3.4633
sai s6
2.1E-04 3.0E-04 1.4E-04 3.7E-05 5.0E-O5 1.6E-05 l.OE-03 :.0E-03
9.5E-04
1.3E-03 2.2E-03 5.8E-04
Trang 36e,.10'
e, 10' e,.IO*
e.lO' e,.10' e.lO'
^
o
^ u
e,.10' e,.10' e.lO'
lan lap 3
16 giai doan 0.7054 0.5448 1.1189 -5.0196 3.0462 1.9735 0.7065 0.7926 1.2556 -6.1502 2.6947 3.4555
30 giai doan 0.7046 0.5447 1.1183 -5.0175 3.0444 1.9731 0.7122 0.7886 1.2514 -6.1339 2.6897 3.4442
lan lap 4
0.7053 0.5447 1.1188 -5.0198 3.0463 1.9735 0.7039 0.7888 1.2522 -6.1496 2.6972 3.4524
0.7045 0.5447 1.1183 -5.0175 3.0445 1.9731 0.7028 0.7894 1.2521 -6.1403 2.6925 3.4478
sai s(5
3 - ^
l,5E-04 2.0E-04 9.4E-05 2.6E-05 3.6E-05 l.lE-05 3.8E-03 4.9E-03 2.7E-03 l.OE-04 9.1E-04 8.9E-04
7.8E-05 9.9E-05 4.8E-05 1.4E-05 1.9E-05 5.2E-06 9.3E-04 l.lE-03 6.1E-04 l.OE-03 l.OE-03 l.OE-03
Trang 37Vi c^c dai lugng Ih h^m cua r vh t , trong dd a <r<bj>0
n6n ta chia bdn kinh b-a th&nh a doan bdi a + 1 di^mchia
Dua v^o c4c dai lugng khdng thu nguyfin :
;r[b -a )a^ ' cr, ' b
Ap dung tinh todn vdi dng tli^p da tdi 40X cd cdc dac trung vat liCu :
• 5 ^ = 0,028 , CT, = 8 5 8 — ^ , chiu tai Uieo 3 c^ch ( hinh IL Li),
Qiia dfeu c^c giai doan tlieo tliani sd tai / tlico ca 3 each dat Iai
Bang II 1.1 cho kft qua tinli toan theo ly tliuyft qud trinli dd cong
trung binli d 3 giai doan tai tlieo cdch dat tai (1) tai di^in r = a, \^
Xitn hinh II 1.2 cho hinh anh deo phftn bd theo dd d^y d hai giai
doan Uiir 22 v& thu 23
Di khao s^t su dn dinh cua phuang phap lap va so s^nli cdc kft qua
tinh toan khi ap dung he thuc qua trinh bie'n dang cd dd cong trung
buih vdi qu^ trinh bi^n dang cd do cong phuc tap , ti^n li5nh tinh loan
vdi tai ngo&i tic dung \ti\ dng vdi 3 each dat tai khac nhau v5 dugc chia
tliSiili cac giai doan dat tai khdc nhau : /i, = n,/!^ = 16,«3 = 30 nhu dugc
nid t^ trfin hinh IL 1.1 Kft qua tinh toin tai cic didin trong tlu^nh cua
dng r=a dugc ghi trong bang ILL2 , tr^n dd sd liCu dugc ghi tli5nli
hai rihdm : nhdm a) dugc tinh tiieo ly tiiuyC^t qud trhih cd dd cong
trung binli, nhdm b) dugc tinh tlieo ly Uiuy^^t qua trinh dftn deo tdng
quit Trdn hinlilLLS md ti dd iJii quy dao bi^n dang Iai nhung di6n
trCn tii^nh trong cua dng r=a
Tfir cic k€{ qua tinh toin , cho plidp nit ra mdt sd nhan x<5t sau :
a) Trong sudt qui trinh, sai sd giua hai l^i lap tliu 7 vi 8 cua cic
dai lugng ung suit, bi^ndang, cudng dd ihig suit d^u nhd lion sai sd
giira hai lltn lap thii* 2 vi 3 cua chiing
Trang 38b) Vdi Cling mdt cich dat tai, khi chia qui trinh dat tai theo sd giai doan tang Ifin thi sai sd giua hai giai doan cua tit ca dai lugng d moi giai doan tai se giam di
c) TInh toin klii ip dung ht tliuc qui trinli dan deo tdng quit cho sai
sd giua hai Ihn lap cua tit ca cic dai lugng trong sudt qua trijih dat tai
nhd han khi ip dung he thuc qui trinh biS^n dang cd dd cong trung binh Tuy nhien vdi qui trinh bi^n dang phuc tap cd dd cong nhd ( nhu trong bii toin dang x^t) thi hai ly thuye^t cho ket qua khi gin nhau, di^u niy
cd nghia li ddi vdi qui trinh loai nhu vSy cd U\i six dung he tliuc xic
dinh cua ly tliuy^ft cd dd cong trung binh da du phan inh ung xu cua ke^l ciu
d) llidi gian tinh toin : Lip trinh bang ngdii ngCr PASCAL, tinh tren niiy AT, vdi 8 lin lap tren mdi giai doan tai ( sd giai doan tilr 20 d6i
3 0 ) : nS'u su dung he thiJc qui trinh din deo tdng quit thi tlidi gian tinli
Mi khoang tilr 60 d^n 90 giiy, ndu su dung he tliiJc qui trinh cd dd
cong trung binh thi hft khoang tit 45 d6i 60 giiy
Qua kft qua tinh toin cua bai toin niy cho tliiy rang : phuang phip bidii th^ nghiem din hdi hdi tu d moi giai doan Tdc d6 li6i tu trong ca qui trinli phu tliudc nlii^u vao muc dd phuc tap cua qua truih dai tai Ddi vdi cic qua trinh dat tai da xet: tdc do hoi tu khi cao, bao dam llnh
6u dinh cua phuang phip, sai sd ciia cac dai lugng nhd di klii sd giai
doan chia tang len
Trang 39$2 T R ^ G T I I A I D A N DEO CUA IIIANH TRON CH}UXOAlVVALl,t^
KfeODOCTRVC
X^t tlianli trdn, dii, bin kinli bang a , bang vit lieu klidng u6\\ dugc, tii b^n tuy^n tinli, chiu luc keo - nen doc true P(t) vi md men xoan M(t)
Su dung gia tliie't bin kinh tliang va tliie't dien phang vi xet bii toin trong he toa dd tru : true z hudng theo tnic cua thanh Do tinh chit ddi
xung cua vit \hi vi luc tic dung, nen cic dai lugng bi^u tlii ung suit vi
bie^n dang khdng phu tliudc vio gdc 6 vi z , cic dai lugng cliuy^n vi
khdng phu tliudc vio gdc 6 Cic Ihinli pliin ung suit chi cdn Iai thinh phin c^ vi ex ^^ 1^ khac klidiig, con cic tlianh phin khdc deu
bang khdng, do viy cic phu^ang trinh cin bang d^u llida man ddng nhit
Goi ^ l i gdc quay giua hai tliiet dien cich nhau mdt dan vj, do gia
thi^t bin kinli tliang vi tliie't dien phang, chuye'n dich ^o ~ P^^^ do
^e = ^ez = fi{nr:e^ = e^{t)\ (11.2.1)
cdn cic dai lugng ^^ = <?^ - 0
Cilia qui trinli dat tai tilr diu d6i tlidi diem dang \6i tliinli // giai
doan Khi dd tai giai doan dang xet ( tliu /;) nghiem cua bii toin dugc giii bang phuang phip gin diing lien ti^p tren ca sd biet dugc nghiem
Trang 40tiy giai doan diu de^n cudi giai doan tliu n-l Tai gin diing thu k a
• giai doan thu* n , lien he giira cic tliinh phin ung suit vi bi6i dang :