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DAI HOC
Qudc GIA
HA
NOI
TRllONG DAI
HOC KHOA HOC
TV
NHI£N
Va
Khic
Bay
TCVH
TOAN
TRANG
THAI
DAN
DEO
CUA MOT
SO
K&V
CAU
emu QUA TRENfH
DAT
TAI
PHtfC TAP
BANG
PHUONG
PHAP
BI^N
rafi

NGHltM
DAN
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
'•y.o\\
H^
Noi - 1995

t
•
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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
MOTSO
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
Oiwang III KHAo
SAT
PHITONG
PHAP
BI€N
Tsi NGHE^M
DAN
nbi
TRONG
MOT
S6
BAI
TOAN V6
$ 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
KH6NG
GIAN
$ 1. Tru tron ngan chiu tai
phiJc
tap 104
$2.
Nhanxet 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
to^
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 tlm^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,
52].
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^
thu^
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
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 :
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
ph^
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
ph^
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
BI€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^
(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
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
(1.1.2)
s - n
Ou
=
0u[Zmis),pis),
Tis')y^_^
(„

=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)
(1.1,3)
ds
da
ds

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

<T^,
0^
se
la
nghiem cua he
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
dao bieh dang . Theo dinh
nghia
ta cd : cos
d,
=
cT.p,
do dd :
d0_

ds sin 0,
d_
ds
\
dp.
ds sin
0,
Dat /
=
-
—»
P\
sin
9^
d
ds
(
^ \
a
rJ
do
ds
XxP:
va
Pi'Cr-
a^cosO^
= Icr^.sin
0^
do
nen tacd:

-^^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.
=
—
, ^ =
X\Pi^
ds ds
ds
XvPv

va
Cling vdi
(1.1.5),
(1.1.6) dan vS
da
ds
—
^f>ctg9\a
a,./
d.
V^u
sin
0
ds
r\
^ I -*
d
3
Do cos
^ =
—.
a.
ds
nen
ta duac
sin
0
¥
_^^u/
cos 6 sin

^
CT.d^
-
—;—cr
(1.1.7)
Lien he
nguge
Iai :
,
-•
sin
^
,
-*
a 3 =
d
(7+
sin 6 cos 6
+
^uf
¥
ad a
a
(1.1.8)
Nffu bi^u di6n dirdi
dang
ten-xa
ta
dirge
:

^.,
=
-
2a,.f
3 sin
e "'^
i
+
W
,
^uf
cos
^
sin 6
§M^_S
(1.1.9)
CT=3Ke
(1.1.10)
Theo [ 61 ]
cdc
h^
/,
f cd
dang
hi^n
sau
dfly
11
/ = -
sin 6

1 +
3Gs
V '^u
-1
(
1 -
COS
6
(1.1.11)
V^=
^'(;y)cos^,-(3G-f(5))[
1 - cos
^
9
=
arccos
—^^-^
CTu^u
(1.1.12)
(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
,
r
3G

.
^
/
=
-—sm^
(1.1.13)
^^=3G.cos^
he thiJc (1.1.9) dfin vS
dinh
luat
Hooke
: ^ ,j = 2Gd
+
(^a
trinh dat
tAi
dan
gian
: Khi dd cd
0
= 0
Lim
<^o
sin^
/ _
1
Lim
ly
=
(h\s)

z:>
(1.1.9) dfin vfe
^.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
'
3f.,
^
e„ +
^
^&
^
^'^u
«^u
y

^u
(1.1.15)
V6i qud
trinh dat tai
dcfn
gian ta co:
^ = ^u;^= ^u =
^u\
da
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
0
2
\
'«-11^»
^
(
r(s)-^
Sue
id^'ki
s
J
<TI
'
(1.1.16)
He

thiJc (1.1.16) la
tdng quat hda he tliuc Prandtl - Reuss cho vat
lieu deo
1^
tudng
va
he
tliiJc
Prager cho vat lieu tai
bSn
.
+
Qak
trinh
cat
tai:
xky ra
chae
chan
klii 0
=
7t , klii
dd :
Lim-^^
;L/m|^=-3G
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]
13
0
3
BAI
TOAN
BI£N
VA
PHirONG
PHAP
BI£N
nii NG1I1$M
DAN
HbicuALTf TSUYh Qvk
TRINH
BI£N
D^NG
DAN DEO
2.1

B^i
toan
bi^n
cua
1^
thuyet
qua trinh
bi^n
dang
d^n
deo.
. 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 :
+ He tliuc
Cauchy
:
+
P^/
=0 , X€Q
(1.2.1)
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^
1 +
f3Gs j]
I
^
J
fl-cosf?^
I
2
J
a
(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
= -
sin
0
r.„
cT

Y1
- cos
6'V
P =
¥
_
cos 6
,V\
(
i'
A(I-COS^)
^ (0-[3G
-
^
{^)y-^,
2^
cos
6*
(1.2.8)
Dat
/l = 3G(l-£y,) ,P = 3G(l-fyJ
khi
d6:
' 3G
I
3.G S
1-
n
- cos
0

(1.2.9)
- = '-i^='-|E
1 +
n
-cos 6'V 1
COS
Q
v^
(1.2.7)
trathanh:
dS,=
2GJ„cJ^,
-
1Ga),5,,5^,
-
3G(ftjj
-
o),)
"
\
a".
de,
(1.2.10)
"
y
Dal
^y«
-
S,.S
2Go)^6,^5,1

+3G(GJ2
- ft),)
'-^
(1.2.11)
»
/
Klii
do
(1.2.10) ti6 thaiili
:
^5.
=
2
Gde,,
+
/?
«c/e^^
(1.2.12)
Do da =
3Kde
v^
dat
;///,«=
T-^^I*^//*/+
^//w (12.13)
heUiiJc (1.2.12)
dUnd^n:
^c^y =(/l<Jy<5«+2GcJ,tJ;,+//y«)j£« (1-2.14)
16
tu.

=
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
/
:
u,{x,i,)
=«/"'
= ur + t
^"
n
m
= 1
^,(^./j =
<"'
=
<'+
2:
AC'

,
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
'^
5r = 2G<> + 2
J^^v.^«^^

(1.2.16)
"=»
Vi
Trong
(1.2.16)
ta udc lugng
tich phSn
'JKne.dt =
i[/?<r"
+
^J^'lA^-"'
=
A„e,^
(1.2.17)
'1 I
17
rt-i
Si"'
=
2Gej;^+X^'^'>j+^ne^
m=\
Tit (1.2.14)
taco:
(1.2.18)
(1.2.19)
Vdi udc lugng
tlch
phan
:
J

tm-l
_
J_
(m-\) (m) (m)
_
Hijki
e,jki
-
^Hijki
'^
Hijki
^
^eu
~
Am £ij
(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,
= sr-[^f
A„5,
+

A<„*-%1.;,,
txfin ^,
«;"•*>
=
^(")
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^)['-(
1 -cos
0^"-^-^
3Gs
)•].
(1.2.26)
fi;<"
-'^^ -
~3G—^^^
"•-^JTZr^J^]
(1.2.
27)

19
(
a ^1
, P>1 )
cos
^c-*-')
=
-^ y
in.k-l) ^^(n.k-l
(1.2.28)
yC",*-!)
_
_y("-') ^
A^^"'*"'
(1.2.29)
vdi
A5"'*-"
=
!(<•""-<"")•(<•""-<-")
1/2
(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-*^
= 2G.J-*UXA , +
A^r»>e
ij (1.2.31)
m
= l
Trong dd ;
20
^V'e,
=
HRiu''
+
^i;/*-"][<'*-'^
- <M (1.2.32)
n-1
D^
dl
sd
dung
sau
nay
ta
k^
hieu
:

^ij"'
-^
^„^tj
"^
^n
^^ij
m=\
va
'*{rt.k-\)
n-\
khi
do
(1.2.24)
v^
(1.2.31)
co
th^
vi^t
CTJ,"'*^ =
X0i"'''^Sy
+
2G4"-*)
+
cT^^"-*^
(1.2.33)
'J
U iJ
(1.2.34)
Difeu
ki6n

hdi tu : Tai
giai doan
tliiJ
n
cdc
dai
lircnig
chuyfi'n
dich
c^i
thoa
man:
i
i
<g
^{n,k)_^{n,k-2)
i i
voi
q<J
(1.2.35)
tircmg tir
nhu
vSy
cho
c^c
dai
lucmg
^."'^\J;"''^\o^"'^\
'
ij ij

u
2.
M6
hinh
tinh
todn
:
Vdi
nCi
dung
cua
phucmg phip
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:=\
3lrt
da,j:
= 0
k:=0
m.
=
m
+
\
Luu kft

qua
giai doan
m
av"" '^:=<-nJa;;
TInh cac dai lugng :
m,k-\)
Sm,k-\)
m,k-\)
(m,k
-I)
m
(/
0
k:=k+l
luih
:
41'^,
'/o-;:=^:'f,
///•////
I.l
22
3.
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
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 :
ch(m.x)-
i./^.
i nen

^ ^
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),JoWv
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
TINH
CHAT CUA PHlTONG

PIlAl'
BI£N
Tfrf
NGHI$M
DAN
HOI
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.
01
TRi^G
IHAI
DAN
Dto
CUA ONG
DAY
CIIJU
AP
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