LYTHUYETTAM CO XET BIEN DANG TRUITT
r/i5. NGUYiN THUVANH Truxyng Cao ding GTVT
Tom tat: Ly thuyit tim chju uon co xet biin dang trwat dwac xay dirng theo cac phwang phap khac nhau, tuy nhien no vin chwa hoan thien. Trong bai bao nay tac gia gi&i thieu phwong phap mai xuat phat tu- viec xem cac ham chuyin vj, ham lire cat trong tim diu la cac ham cin xac dinh di xay dwng ly thuyit tim xet biin dang trwat. V&i each lam nay, fy thuyit tim xet biin dang trwgt cOng dong th&i la ly thuyit tim cho phep thoa man ba diiu kien bien.
Brief: Theory of bending ptate with shear strain is formed by different methods; however, it is not perfect. In this article, the author has introduced a new method deriving from consideration of dfs- ptacement function and shear function in the ptate, which must be determined for formation of theory of plate with shear strain. By doing so, theory of plate with shear strain concurrently means the theory of plate which enables to satisfy three end conditions.
V
I n de xet bien dang trugt trong t i m chju uon khong phai la v i n d l mai va da dugc cac nha khoa hgc nghien CLPU kha ky va dugc trinh bay trong cac tai lieu nhu [1,2] va trong cac tai lieu v l phuang phap tinh [3,4]. Dac diem ehung cua cac phuang phap nay la deu din bai toan t i m chiu uln v l mgt phuang trinh vi phan d p bin d i i vai do vong. Cach xay dung bai toan nhu vay la chua hoan chinh va do do dan din kho khan trong khi giai ngay ca khi dung phuang phap phin tCp huu han [3,4]. Trong bai bao nay tac gia trinh bay phuang phap xet bien dang trugt trong t i m vai viec xem ham do vong W, Qx, Qy la cac ham can tim, xay dung bai toan theo phuang phap nguyen ly cue trj Gauss, SLP dung phep tinh biin phan dua ra phuang trinh vi phan can blng, cac phuang trinh vi phan lien he giua momen va luc d t va dac biet la dua ra cac diiu kien bien cua t i m khi xet biin dang trugt.Van SLP dung cac gia thiet cua ly thuyit tam co dien (ly thuyit tim Kierchhoff) va xet tam co chieu day h khong doi tren b l mat cua no. Ggi W(x,y) la ham do vong va Qx(x,y), Qy (x,y) la cac ham luc cat thi goc xoay tai mat trung binh do luc d t gay ra se dugc tfnh bang:
Yx =
Gh
Tv =_^Qy
Gh
(1)Trong do h la chiiu day t i m , G la mo-dun trugt cua vat lieu, a la he s i xet ipng s u i t d t max a mat trung binh (khi xet bien dang trugt thuang l l y a=1.2 [1]). Og CLPng (chong) d t Gh dugc xac djnh
qua do cung uon D cua tam nhu sau:
Eh ^ ^ Eh' oD^
2(1+v)' 1 2 ( 1 - V ' ) ' G A "
Gh=-
5(1-V) (2)v: la he so Poisson cua vat lieu.
Theo ly thuyit tam xet bien dang trugt cua Timoshenko [1], goc xoay px, Py do cac mo men Mx, My tuang LPng gay ra se b l n g do d i e eua mat vong trip goc xoay do luc d t gay ra nen ta co:
' Py ~
Px =
dx Gh dy Gh (3)
Nhu vay, tiet dien tam truac khi bien dang la phang va t h i n g goc vai mat trung binh thi sau khi bien dang van phang nhung do xet bien dang trugt thi khong con t h i n g goc vai mat trung binh nua.
Cac b i i n dang uon '^-^, ^>'va xoan ^^J'{cac6g cong cua duang dan hii) cua tam do v i n bao dam gia thiit t i l t dien phang nen dugc xac djnh nhu sau:
ap, _ d'w^a a a
X
x>
dx' d'W
A/XV
dx
dy dy'
5P. , 3P;
1 + -
dy dx
Gh dx
a dQ^
Gh dy d'W
(4)
a ( ^ .
<*e,,,
dxdy 2Gh. dy dx
Ngi luc momen trong tam dugc xac djnh bang:
M,=D(x,+\iXy), My=^(Xy+^X.),
M^=D(l-[i)X
V(5) TAP CHi CAU DU&NG VIET NAM
KHOA HOC-CONG NGHE
Dua (4) vao (5) co cac cong thuc:
M =D , d'W a dQ , , d'W ^ a dQ
M..=D
dx' ' d'W
Gh dx
dy M=D{l-v)
d/
. / d"W ) + v ( — ^ +
Gh dy
a aa
Xet tim chu nhat co chiiu dai cac canh la a va b (hinh ve). Dieu kien dung cua phiem ham Z duac viet nhu sau:
Gh dy
d'w a ,aa , dQ,
dx' Gh dx
5Z=JJM,6
r 7^2d'W dy
\dx' dx
a b
dxdy+ jJM^d
-(- • + -
/ :^2
(6) dxdy 2Gh dy dx Co the luu y rang cac cong thuc (6) vai a =1.2 cung la cac cong thuc xac djnh momen uln va momen xoln trong tam theo ly thuyet tim xet biin dang trugt cua Reissner [1]. Khi khong xet bien dang trugt, G —> °°, cac cong thCpc (6) la cac cong thuc xac djnh cac ngi luc momen trong tim theo ly thuyit t i m thong thuang (ly thuyet tam Kierchhoff).
Biit dugc cac quan he (4) va (5) de dang xay dung bai toan tfnh tim co xet biin dang trugt theo phuang phap nguyen ly cue trj Gauss. De xac djnh, xet tim chju lue phan bo q . Lugng cuang buc cua bai toan viit nhu sau:
• ^ — ^Mx '^'^My + ' ^ M x y '^^Qx,Qy ^ ^ - P " > ^^f^
d'W ^dy^^
dy' dy
a b
dxdy+2\\Mj>
0 0
d'w + dxdy dxdy +1J ( a s Y, + QP y)dxdy
00
jjqbWdxdy =0 (8)
2 ay dx
a b
Phugng trinh (8) co ba dai lugng biin phan khac nhau la Sff, Sy, va 5y^ nen co the tach thanh ba phuang trinh sau:
d'W^
\\MM--r dxdy^\\M^h
2 I T / ^
d'w
dx' '' ii ^ \ dy'
J a b ( Ti^
dxdy-v
0 0
d'w
V dxdy I
\ a b
\dxdy -\\ q5Wdxdy = 0 (9)
d'w
• +dx' dx d'w dy^
)dQ.
- + •
dy' dy -)dQ.
n n Z^^=\M^1yd^ = \M^{-
Q. a
Z^ =2fM Y da = 2\M
Mxy J xyA^xy J xy
n Q
d'w 1
3Y 3Ydxdy 2 dy dx \ Q_
Ze.,ey=\hy.+Qyyy]d^ ,Z,=-\qWdQ. (7)
a Q.
Trong cac bieu thuc tren Q la dien tfch tim. Khi tim cue tieu cua Z d n xem cac biin dang uln %„X^ va Z^
la dai lugng biin phan dgc lap dii vai M, M, va M^, cac bien dang trugt Y.va % la cac dai lugng biin phan dgc lap doi vai eae luc d t a va g , ehuyen vj W la dai lugng bien
^ phan doe lap dii vai luc tae dung<7.
00 V / 00 t, \ -^
dxdy
+
0 0
J J M ^ 5 | ^ W + 2jjM^5
y^y) 0 0
0 0
]]Q^^^dxdy = Q (10) 1 ay "i
dxdy
2 ax J ^
\\Qy^lydxdy = Q (11)
a b
De CO the vua nhan dugc phuang trinh vi phan can bang, vCpa nhan dugc cac diiu kien bien cua tim ta tfnh tfch phan tung phan cac phuang trinh (9),(10)va(11).
Tfch phan tung phan cac thanh phin cua
b
phuang trinh (9)
\\Mb
0 0 V
dx'
o
dxdy = -\M^
J
dw
dx dy
a
(9.1)
+ J ^|5^|o <^y-\\ —hWdxdy
^ 0 0 ox 0 dx
(9.2) (9.3)
TAP CHi CAU omOTNG VIET NAM
a b ri'W "
]\Mbi-^)dxdy = -\M^
0 0 g y 0
dW
(9.4)
+ '-bW\dx-\
J :i,, \ lo J J 0 dy 0 0
d/
dx dy
-bWdxdy
(9.5) (9.6) Thanh phin thCp 3 (chLPa momen xoan) cua phuang trinh (9) dugc tach ra nhu sau:
a b
d'w.
n^M-^->''"y=-i^' dW dy dy
(9.7)
^f-M^\bW\'dx-rf-^6Wdxdy { dx ' '° ii dxdy
(9.8) )(9.9
dw
dx dx
a b ri'W "
\]M^bi---)dxdy = -]M^^
0 0 oxdy 0 (9.10)
+ '-\m\ dy-\{ '-dWdxdy
0 dy 0 0 0 dxdy
(9.11) (9.12) Gop cae bieu thCpc (9.3),(9.6),(9.9),(9.12) va thanh phin culi cua phuang trinh (9) (thanh phan chua ngoai luc), viet dugc:
bang cua tim khi xet bien dang trugt. Trong truang hgp khong xet biin dang trugt, G^<>°, hoac chu y tai (2), h=0, phuang trinh (13) tra thanh phuang trinh vi phan xac djnh do vong cua tam cua ly thuyet tim Kierchhoff.
Tfch phan tCpnq phin cac thanh phan cua phuang trinh (10)
]]Mb(^)dxdy =
00 OX^
K|6Yj^j_n^5Y^x4^
0, 0 0 0 dx
(10.1) (10.2)
2]]Mj{\^)dxdy =
j^^lSyJ dx -\\—^dyJxdy
0 0 0 0 o y
(10.3) (10.4) Gop (10.2),(10.4) va thanh phin euli eua phuang trinh (10), viit dugc:
w
a b 0 0dM dM^
dx dy + a hyJxdy^Q (10.5)
Vi bien phan dj^ la bat kl nen tu (10.5) nhan duac:^ dM dM^
Q.=^ + ^ (14) dx ay
\.d'M
•JJ( dx'
d'M d'M
+ 2 — - ^ + -dxdy dy'
+ q)hWdxdy-0 Thay (6) vao phugng trinh (14) taco:
(9.13) VI bien phan bW\a bit ky, tCp (9.13) ta co:
a x 3X .f^. ^_o
Q ^ ^djAW) ^ h' 3 ^ ^h'd'Q^
dx'
dy' dxdy (12)Dua (6) vao phuang trinh (12) nhan dugc:
dx' d/ ' "•'• dx' d/ (13) Phuang trinh (13) la phuang trinh vi phan can
dx 5(1-V) dx' 10 dy' h'l+vd%
-\- ^ (15)
101-vaxay;
,,^, d'w d'w
Khi khong xet biin dang trugt, h=0, phuang trinh (15) tra thanh cong thuc xac djnh luc d t Qx cua ly thuyit tim thong thuang.
Tfch phan tuna phan cae thanh phln_rMa phuang trinh (11)
TAP CHi CAU DUDNG VIET NAM
KHOA HOC-CONG NGHE
jJM^d -^dxdy = JM^ |6Y^ I dx - jj^dy^dxdy
0 0 oy - ~0 ""'•'
(11.1)
0 0 dy (11.2)
2JJM^5-^dxdy = JM^py,\ dy-jj—f8y^dxdy
00 -^ ox 0 0 0 0 ox
(11.3) (11.4)
Gop (11.2),(11.4) va thanh phin culi cua phugng trinh (11), viit dugc:
a o
W
0 0
dM dM
dy dx +a 6Y dxdy = 0 (11.5)
VI bien phan djy la bit ki nen ti> (11.5) ta co:
a = dM dM dy dx + (16)
Dua (6) vao (16) nhan dugc phuang trinh sau:
Q = - ^ M ^ . h' a u h'd'Q^
+
dy 5(1-V) dy' AM+v d'Q^
101-v dxdy
+
10 dr' IX(17)
Khi khong xet biin dang trugt, h=0, phuang trinh (17) trg thanh eong thuc xac djnh luc d t Qy eua ly thuyit tam thong thuang.
Bang each tinh biin phan neu tren nhan duge ba phuang trinh (13,(15) va (17) dexac djnh ba ham chua biit la W(x,y),Qx(x,y) va Qy(x,y). Khi khong xet biin dang trugt, cac phuang trinh nay trg thanh phuang trinh vi phan dii vai do vong va cac eong thue xac djnh luc d t cua tim cua ly thuyit thong thuang.
Bay gia xet cac diiu kien bien.
Canhx=Ovax=a
Gop hai phuang trinh (9.1) va (10.1) ta co:
S(-^r- + YJ
JM^ dx dy = 0
(a) Phuang trinh (a) dugc thoa man khi:Canh CO lien kit khap
. . . ... d'w a dQ^ d'w a dQ
dx' Gh dx dy' Gh dy
(18)
Canh CO lien ket ngam
dx Gh
(19)
TAP CHi CAU DUONG VIET NAM
Gop hai phuang trinh (9.2) va (9.11), chu y tai phuang trinh (14), ta co:
0 dx UJ 0
Phuang trinh (b) thoa man khi:
Canh CO goi tua W=0
Canh tu do (khong co gii tua) Qx = 0
Gop hai phuang trinh (9.7) va (11.3), ta co:
\M
dw a
^ dy Gh^'^ dy = 0
(b)
(20) (21)
(c) Phuang trinh (c) dugc thoa man khi:
dy Gh^'
Hoac khi:
/
M =0^ d'w a ,aa ^Q.
+
(- + •dxdy 2Gh dy dx
-) = 0(22) De bao dam cho momen xoan bang khong a mep tim, ta chgn diiu kien (22).
Nhu vay, ly thuyit xet biin dang trugt vai viec xem cac ham do vong va luc d t la cac ham can xac dinh cho phep thoa man ba dieu kien bien tren moi canh tim. Do la cac dieu kien v l momen uon, luc d t va momen xoan hoac v l goc xoay do momen Mx va chuyen vj.
Blng each eho biin dang trugt trong cac phuang trinh (a),(b) va (c) bang khong d l dang nhan dugc cac dieu kien bien cua ly thuyit t i m Kierchhgff. That vay, phugng trinh (a) bay gia co dang:
M 6(-^)
ox dy = 0
(d) Phuang trinh (d) dugc thoa man khi:Canh CO lien kit khgp
d'w d'w
M ^ = O ^ Z ) ( - V f - V ? r ^ ) = 0 (18a)
dx' dy'
Canh CO lien kit ngam dW
dx
= 0 (19a)(Chu y, dieu kien (19.a) khac vgi (19)).
Phugng trinh (b) va phuang trinh (c) khi khong xet biin dang trugt co dang:
J(
udM, dM^ dx dy
+ •-)\W\]dy = ]Q:pwldy = 0
\M
S(- dw
dy dy = 0
(e) (f) Bai vl chl con mgt diiu kien bien nen d n kit hgp hai phuang trinh (e) va (f). Tfch phan tung phin phuang trinh (f):
\M
5(- dW_
17 dy=-M IdWl" +
•^ ^ I lo 0
\—^\Widy = Q (g)
0 dy
Gop hai phuang trinh (e) va (g), ta co:
j(a
+ •dM
dx -)\hW\[dy = 0,
M \\WY
^ I lo
= 0
Phuang trinh (h) thoa man khi:
Canh CO goi tua W=0
Canh tu do (khong eo goi tua)
a + dM^ dy
= 0 - ^d'w
dx' + (2-v)
(h)
(23)
(20a)
d'w
dxdy'
= 0(21a) Cuoi cung, dieu kien (23) chi ra ring niu goc tim CO chuyin vj thi momen xoan tai goc phai bangkhong.
Ly thuyit tim Kierchhoff khong co dieu kien (22).
SLP dung cac thanh phan con lai cua tfch phan tung phan neu tren de xay dung cac diiu kien bien tai cac canh y=0 va y=b cung nhan dugc kit qua tugng t u , khong d n thiit trinh bay lai g day.
Ketluan
Ly thuyit tim xet bien dang trugt trinh bay g tren vin dung ba gia thiet cua iy thuyit tim Kierchhoff. Khi xet biin dang trugt thi tilt dien truac khi bien dang la phang va thang goc vai mat trung hoa, sau khi biin dang vin phang nhung khong con thing goc vai mat trung hoa nu'a.
Phuang phap xem hai ham luc d t Qx , Qy va ham do vong la ba ham an d n xac djnh de xet biin dang trugt la phuang phap mai. Phuang phap nay cung da dugc dung dii vai dim [5].
Xay dung bai toan theo phuang phap nguyen ly cue trj Gauss va SLP dung phep tfnh biin phan nhan dugc ba phuang triph vi phan de xac djnh ba ham chua biit va cac dieu kien bien cua tim (trong bai xet tim chu nhat). Diiu dac biet la do dung them an nen co them diiu kien bien la momen xoan phai bang khong tren moi canh tim.
Ly thuyit tim xet biin dang trugt trinh bay a tren xem ly thuyit tam Kierchhoff la truang hgp riengB
TAI Lieu THAM K H A O :
[1] KP.Timosenkd - X.V6in6pxki-Krige (1971), Tdm vd vd
[2] B.A.KuceAe6 (1973), PACHET TIAACTMH, MOCKBA, CTP0MM3MT.
[3] Klaus - Jurgen Bathe (1996), Finite Element Procedures, Prentice
[4] 0.CZienkiewicz and R.L.Taylor, The Finite Element Method.
[5] VU Thanh Thiiy, xdy dimg bdi todn ddm khi xet day dil 2 ngi Ivcc momen udn vd life cdt, Tqp chi Xdy dimg thdng 1-2009
TAP CHi CAU DUlorNG VIET NAM
