(Luận án tiến sĩ) tính toán trạng thái đàn dẻo của một số kết cấu chụi quá trình đặt tải phức tạp bằng phương pháp biến thể nghiệm đàn hồi luận án PTS cơ học1 02 21

^

DAI 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!)ujMa 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
'ãy.o\\

H^ Noi - 1995
t

ã

- - ^ .

^ô*w


MUC

•

LUC
a

MyCLUC
MO DXU

1

Oiirangl cAc H $ TEFDC CO BAN CXJALYTHUY^QUA TRINH BI€N DANG
DAN DEC VA PHiroNG PHAP BE6J TH£ 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
Qiirang IL KHAO SATPHUONG PHAP BI6V Trf

NGHI$M DAN HOI
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
$ 2. Trang thdi dan deo thanh tron chiu xoan va lire keo doc true

$ 3. Trang thai dan deo cua ban trdn mat chiu lire phap va bi6n
chiu keo nen d^i xiitig
$4. Nh$n xet cac k^t qua trong churofng II

23
37
46
58

Oiwang III KHAo SAT PHITONG PHAP BI€N Tsi NGHE^M DAN nbi
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
Chirang IV

KHAO SAT PHITONG PHAP BI£N TH^ NGHI$M DAN HOI
TRONG BAI TOAN K H 6 N G GIAN

$ 1. Tru tron ngan chiu tai phiJc tap
$2. Nhanxet

104
124

KftT LUAN

125

TAI U&V IHAM KHAO

127

PHULUC


MO

DAU

Nghien 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


man: 0^ a)(€j<, a}{ej + — e „ < l cdc t^c gia da chi ra rang : day c^c
nghiem 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
TieJ = 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 :

g(^. ,r,/;)^^^°^"-^(^-^'^\ va ^(r,/^)= ^^I^^ii^
3G,e,
-' ' '
K,
v6i Go.ATo la m6 dun tnrot, n6n th^ tfch cua vat lieu khi khOng c6 buc
xa va 6 nhiet d6 T^
Phuang phap tham s6 dan hbi thay d6i [ 5 ]: Trong phuang phap
nay cac he thitc cua 15^ thuyet deo dugc vift dudi dang dinh luat Hooke
ma rOng, cac he s6 trong do phu thu6c vao trang thai ung suat, bie^n
dang. O g ^ diing dang x6t, cac lie s(5 nay dugc l^y liico gia tri tilr
gan dung tnidc. Nhu vay bai todn dua vS viec giai lien ti^p cac bai
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


hgp v(Ji phuang phap gan diing lien tifi^p [11,42, 51, 54, 57,58} va
phuang phap phan tiJf bien d^ giai cac bai toan cua ly thuyft deo [53].
Nghien 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.


Do tinh chat qui trinh cua ly thuyet nay ma cac he thitc ca ban cua
no iai cang phiJc tap. D^ giai cac bai toan bien cua ly thuyet qua trinh
dan 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.


N6i dung luan an gbm 4 chuang, phan md dau va k^t luan.

PbaatD&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.


cac ktt qua ca ban cua luan an da dugc bao cao tai cac Hdi nghi
khoa hoc va Xemina: Xemina toan nganh Ca hoc vat ran bie'n dang;
Xemina 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.


CHITONG I :
CAC H$ THtrC CO BAN CUA LY THUYfiT QUA
TRINH JM£N DANG DAN DEO vA PHl/ONG PIlAP
B I € N THfe NGHI$M DAN HOI
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.
$ 1. CAC 11$ Tlltlt CO BAN MO TA. LY 11IUY£T QUA TRINII
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^
cos 9„ cos ^„ = 1
—•

(1.1.1)
( 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


8
vdc 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 dd, doan nay dugc goi la
\A cham tr§ //. Do vay a^, 6„ Ih phi^m ham cua dd cong, dd xoan x^
cua quy dao bie^n dang, cua dd dai cung s cua quy dao bi6i dang va
cdc ham vd hudng dac trung cho su thay ddi cac difeu kien vM 1^ cua
qua trinh. Cac dai lugng nay bat bie^n vdi ph^p quay va ph^p chi^u, tuc
la ta cd :
^u = ^u[Zm(s).P(s)J(s)y
s-n

Ou = 0u[Zmis),pis),

Tis')y^_^

(1.1.2)

(„ =1, 5 ; m = 1, 4 )

trong dd s -dd dai cung quy dao tuc tlidi cua quy dao bifn dang s,p
- ap suat, T-nhiet dd.
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, ddi hdi phai cd nhung nghien cuu
tie'p theo. 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 Ihie't 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 dd thay ddi cua cic goc chi phnang cua cic vec ta ung suit
trong ri'pe tu nhiin cua quy dao bi6h dang —j^ vi tdc dd thay ddi
circmg dd ihig suSit ~^ li him ciia cicgii tri tiic th&i cua O^.a^ , dd

as
cong vi dd dii quy dao bidh dang:
2-=/„(^jt,C7„,/^,5)

ds
da
=
ds

y/{ei,,(7,,Zp.s)

(1.1,3)


cos ^„ COS ^„ = 1

( k = 1, 5 ; n = 1. 5 ; p = 1,4)

vdi 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 phuang trinh vi phan thudng phi tuye^n (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 ddng phang cua
vdc ta ling suit, vec tagia sdiing suit, vec ta tdc dd 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 O^.p^)
-*

hay

—>

->

a = ^^(cos O.p^-^ sin 6.P2)

(1.1-4)

vdi 9= 0\ la gdc ti^p can cua vec ta ung suat vdi ti^p iuy€n quy
cos d, = cT.p,

dao bieh dang . Theo dinh nghia ta cd :
do dd :
\

d0_

d_
sin 0, ds

ds

—»

Dat

va

/ =-

d
P\
sin 9^ ds

dp.
ds

sin 0,

( ^ \

a

rJ

do

ds

XxP:

Pi'Cr- a^cosO^ = Icr^.sin 0^

nen tacd:

do

-^^f{0,a^,s)^Xx

(1.1.5)

trong dd ham / , theo gia tliift xac dinli dia phuang, klidng phu thudc
hi^n vao dd 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


10
\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^ :
da"-=
ds

yf{e,(j,,s)

(1.1.6)

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 .
—»

Dao ham (1.1.4) theo s va chu ^ rang p. = —
ds

da
ds

va

XvPv

ds

Cling

vdi

(1.1.5),

(1.1.6)

—•

, --^ = X\Pi^
ds

dan

vS

a , . / d.
^f>ctg9\a--.
sin 0 ds

—
V^u

r\
^
I -* d 3
Do cos ^ = — . a. ds

sin 0

nen ta duac
¥ _ ^ ^ u / CT.d^ cos 6 sin ^ —;—cr

(1.1.7)

Lien he nguge Iai :

, -•
a 3 =

sin ^ , -*
d (7+

sin 6
^uf

cos 6
+

ad

¥

a

a
(1.1.8)

Nffu bi^u di6n dirdi dang ten-xa ta dirge :

^., =

-

2a,.f
i.. + W , ^uf §M^_S
3 sin e "'^
cos ^ sin 6

CT=3Ke
Theo [ 61 ] cdc h ^

(1.1.9)

(1.1.10)
/ , f cd dang hi^n sau dfly


11

/

= -

sin 6 1 + 3Gs
-1
V

'^u

1 - COS 6

(1.1.11)

(

1 - cos ^

V^= ^'(;y)cos^,-(3G-f(5))[

(1.1.12)

9 = arccos —^^-^
CTu^u

( a > l , ^ > l , 0 < < 9 < ; r )

Sy , e^j

la cac thanh phan cua ten - xa lech ung suat va ten - xa lech

bie^n dang. Chii ^ rang cac ham / , V^ d day dfeu su dung cho ca qua
trinh bieh dang chu ddng va bi ddng .

• Cac trudmg hcjrp rl6ng:
+ QU& trinh biffn dang dan hdi:
(T^ = 3Gs , ^ ' ( 5 ) = 3G ,

3G . ^
/r = -—sm^
he thiJc (1.1.9)

(1.1.13)

^^=3G.cos^

dfin vS dinh luat Hooke : ^ ,j =

2Gd

+ (^a trinh dat tAi dan gian : Khi dd cd 0 = 0
Lim /

_ 1

Lim

ly = (h\s)

z:>

(1.1.9)

dfin vfe

<^o sin^

^.4^,,.(^.(.)-^]|..

(1.1,14)

Mat khac, theo 1^ thuyft bie^n dang dan deo nhd cho qua trinli dat tAi
don gian ta cd :


12

Stj = 3 - ^ ^ . /

^.=

'

, hayl&

2 a,.. f
e„ +

3f., ^

^

^&

(1.1.15)

«^u y ^ u

^ ^'^u

V6i qud trinh dat tai dcfn gian ta co:

da..
^ =

^u;^=

^u =

^u\

da
ds

ds

"-= r(s)

thay 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 :
I .
e
/ = — sin 6 ^ —\ y/ = ^'(5)cos 0 ^ (ff'{s)
s
s
Tliay vao(I.1.9) ta dugc
r ( s ) - ^
s J

'«-11^»

^(

02

Sue
id^'ki

\

(1.1.16)
'

He thiJc (1.1.16) la tdng quat hda he tliuc Prandtl - Reuss
lieu deo 1^ tudng va he tliiJc Prager cho vat lieu tai bSn .

cho vat

+ Qak trinh cat tai: xky ra chae chan klii 0 = 7t , klii dd :
Lim-^^
^ * sin p

;L/m|^=-3G
(J.

<'-* *

tilr (1.1.9) dan v^ :

^,

^2Gi,,
-'

"

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]


13
03

DAN

2.1

PHirONG

BAI TOAN BI£N VA

HbicuALTf TSUYh

P H A P BI£N

Qvk

n i i NG1I1$M

TRINH BI£N D^NG DAN DEO

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,^

vdi Q = Q^S ) trong mi^n Q va vdi moi i e
phuang trinli sau day :

+ Phuang trinh can bang :

+ P^/ = 0

,

eC'{Q)r^C\Q)
[ 0, T ] tlida man he

X€Q

(1.2.1)

+ He tliuc Cauchy :
1
'•^

2

(

—^+
\ ^ j

^

dx

X e Q

(1.2.2)

+ Phuang trinh xac dinli:

3 sin 0 "

¥
COS 0 + sin 0

(1.2.3)

cT=3Ke
trong dd :
/ = -

sin^

f3Gs j] fl-cosf?^ a

1+

I ^ JI 2 J

(1.2.4)


14

y/ ^

<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)


15

trong dd : A = -

P =

Dat

¥
_
cos 6

r.„

cT.. Y 1 - cos 6'V

sin 0
, V \

(

i'

A(I-COS^)

^ (0-[3G - ^ {^)y-^,2 ^

/l = 3G(l-£y,)

cos 6*

(1.2.8)

,P = 3 G ( l - f y J

khi d6:

'

3G

I

3.G..S

- = '-i^='-|E

1-

n - cos 0

1+

(1.2.9)

n -cos 6'V

1
COS Q

v^ (1.2.7) trathanh:

a".
"

\

Dal

de,

2GJ„cJ^, - 1Ga),5,,5^, - 3G(ftjj - o),) "

dS,=

^y« -

y

S,.S
2Go)^6,^5,1 + 3 G ( G J 2 - ft),) -'^
»

Klii do

Do da

(1.2.10) ti6 thaiili :

= 3Kde

(1.2.10)

(1.2.11)
/

^ 5 . = 2 Gde,, + /? «c/e^^

v^ dat ; / / / , « = T - ^ ^ I * ^ / / * / + ^//w

(1.2.12)

(12.13)

heUiiJc (1.2.12) dUnd^n:
^c^y =(/l

(1-2.14)


16

tu. = 1 - — ^
3.C/ ,S

, CO, = I-

-5^-y^
3. U

(1.2.15)

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 / :

u,{x,i,) =«/"' = ur + t ^"
n

m= 1

^ , ( ^ . / j = <"' = < ' + 2: A C ' ,

m= 1

^.(^.'J = < ' = < ' + I A < ' ,
m= 1

trong do:

Auj"^ = u,{x ,t„) - u,{x

,t„_^)

Kill do tilr (1.2.12) tacd tli^vi^t
n

'^

5 r = 2G<> + 2 J^^v.^«^^

(1.2.16)

"=» V i

Trong (1.2.16) ta udc lugng tich phSn

'JKne.dt
'1..I

= i [ / ? < r " + ^J^'lA^-"' = A„e,^

(1.2.17)

17
rt-i

Si"' =

2Gej;^+X^'^'>j+^ne^

(1.2.18)

m=\

Tit (1.2.14) t a c o :
(1.2.19)
Vdi udc lugng tlch phan :

J

Hijki e,jki

_ J_
(m-\)
(m)
(m) _
- ^Hijki
'^ Hijki ^ ^eu
~ Am £ij

tm-l

(1.2.20)
Nhirvay tilr (1.2.19) ta diroc :
n-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 :

dx

(1.2.22)


18

(".*) _

—(—
2

dx,

+ —=!
dx,

)

(1.2.23)

m=\

Vol diSu kien bidn :
[^e*^5, + 2G^|;'*)]n, = s r - [ ^ f A„5, + A<„*-%1.;,, txfin ^,

ô;"ã*>

=

^(")

tren

5

(1.2.25)

trong do :

= -[//,]-" + / / , ; - - ' ] [ < - - " - 4 - " j
Vdi

C(n.A-I)o(n.A-I)

3G(a;^"**-'^-6;["-*-»>)

(rt.A-1) i2

[err-"]
(n.A-l)

ft;

.'"•*"•'=('-^s^)['-(
3Gs

fi;<"

1 -cos 0^"-^-^

) • ] . (1.2.26)

-'^^ - ~3G—^^^ "•-^JTZr^J^]

(1.2. 27)


19

(

a ^1 , P>1 )

cos ^c-*-') = -^

y

in.k-l)

(1.2.28)

^^(n.k-l

yC",*-!) _ _y("-') ^ A^^"'*"'

(1.2.29)

1/2

vdi

A5"'*-" =

!

(

<

•

"

"

-

<

"

"

)

•

(

<

•

"

"

-

<

-

"

)

(1.2.30)

Tai lan lap tliu A- = 0 lay A % = 0 . Nhu vay d nidi l;\ii lap lliir k
trong giai doan tliu n ta phai giai bai loan dan hdi tuyen llnh vdi luc
klidi 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
doan thir /? ta cd :
rt-i

6\J-*^ = 2 G . J - * U X A . . ,
m=l

Trong dd ;

+ A^r»>eij

(1.2.31)


20

^V'e,

= HRiu'' + ^i;/*-"][<'*-'^ -
(1.2.32)

n-1

D^ dl sd dung sau nay ta k^ hieu :

^ij"'

- ^ ^„^tj "^ ^n ^^ij
m=\

n-\

'*{rt.k-\)

va

khi do (1.2.24) v^ (1.2.31) co th^ vi^t

CTJ,"'*^ = X0i"'''^Sy

'J

U

+ 2 G 4 " - * ) + cT^^"-*^

(1.2.33)

(1.2.34)

iJ

Difeu ki6n hdi tu : Tai giai doan tliiJ n cdc dai lircnig chuyfi'n dich c ^ i
thoa m a n :

i

i

^{n,k)_^{n,k-2)
i

tircmg tir nhu vSy cho c^c dai lucmg
'

voi q
(1.2.35)

^."'^\J;"''^\o^"'^\..
ij

ij

u

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 ;


21

Dal dau

m:=\

da,j: = 0

av""--'^:=<-nJa;;

3lrt

k:=0

TInh cac dai lugng :
m,k-\)
m,k-\)

m. = m + \

m

Luu kft qua
giai doan m

(/

Sm,k-\)
(m,k - I )

0

k:=k+l

luih : 4 1 ' ^ ,

'/o-;:=^:'f,

///•//// I.l


22

3. MOt s6 diem can ch(i ^ khi tinh to^n:
A) Do lay difeu kien
khi O'u^^s

cr„ ^ CT, ta ap dung quy luat deo d^ tfnh toan,

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^^ 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
mtx

cho tinh x (trong dd Q^^ la gia tri Idn nhat cua a trong klioang
dang xdt) . Vl du klii tinh toan gap cac dai lugng : G^.ch{m.X)
vdi J:G[0,1] va m =1,2,3,...,40. Dift chac rang gia tri cua dai
lugng ^„.c/7(/77.x) Id hiru han, nimig khi m=40, x=J llii gia tri
ch(40)

la qua Idn , khi dd dat

^ ^ c/;(40) ^

) ^ ^-80

: ch(m.x)-

i./^. i

nen

^ 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.


23
CHlTONG n
KHAO SAT CAC T I N H CHAT CUA PHlTONG PIlAl' B I £ 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.
0 1 TRi^G I H A I DAN D t o CUA ONG DAY CIIJU A P L\X:
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