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An asymptotic investigation of the dynamics of eccentrically reinforced plates
~.App~MathsMech~VoL 61, No. 2, pp. 329-331, 1997 © 1997 Elsevier Sdence Ltd All rights reserved. Printed in Great Britain oo21-,s928/97 $24.00+0.00
Pergamon PH: S0021-8928(97)00041.5
AN ASYMPTOTIC INVESTIGATION OF THE DYNAMICS OF ECCENTRICALLY REINFORCED PLATES? I. V. A N D R I A N O V
a n d V. I. P I S K U N O V
Dnepropetrovsk (Rece/ved 9 June 1995) With the aim of extending earlier results [1-9] obtained by the method of avere~ng [5-9], the equations of the vibrations of a plate reinforced by elm;tic n"es mounted e~ontrically with respect to its median surface are investigated. A system of recun~nce equations is derived, o:[ which the first approximation corresponds to the construedve orthotropie theory. It is shown that even this approximation reveals the singularities of asymmetricallydistn~mutedn'bs. Corrections for discretely situated ribs are obtained. © 1997 Elsevier Science Ltd. All rights reserved.
In dimensionless variables, the initial relations describing free vibrations of an eccentrically reinforced plate have the form
I++2
°34
a "~
(1)
=o
BotU~ + un,n, + (VBo, + l ) v gn, = u ~
(2)
Bo,v ~ +v n,~, + (VBo, + 1)u~n, =v
(3)
(w, w,1', u, v ) - = ( w ,
0)
w,1,, u, v )+
(5)
W-lTllql -- Wvhqt = EO~kw~lI
W,l,n,nL -
_
%, - %, = [~u~ - Y2u,, +
vnt -
(6)
n,n,n,= £Otk wv,~,[~,+ Yuv,~,~,+ ~lWv¢
W +
klw~~
(7)
n, = c q v ~ - B o l k z v ~
(s)
_ U +
w=wnl = u = v = 0 w=w~=u=v
=0
11,=~
for
for ~ = 0 ,
u¢Y~a(q)-i)=O
(9) (lO)
L=LI/(2L2)
for ~ = O ,
(11)
L
Here
, ( .o
.Bo. ~ ,
Gogg
_ Ec___SS
;k.2 pcF
EcF
E¢S
B Bol---~,
Eh 8=1_v2
kl = 2L2PcFB° , D = Bh2
DPoh E~J
E~Jj
12
Eh
2L~poh' ~'' = 4-~o~' '~' = 8-b-4' '~= Ob' B o - - -2 ( l + v ) (...)(-) = l i r a ( . . . ) , k = 0 + l y-.~/~_) +0
a2
..... + ( N + I ) / 2
a2
a2
,~ =,~a ~, a' = a--~-+ a-7. (...)~,. _=a--~y(... ) ?Prik£ Mat MeldL 3/hi. 61, No. 2, pp. 341-344, 1997. 329
330
I.V. Andrianov and V. I. Piskunov
Po is the density of the plate material per unit area, Pc is the density per unit length of the n~o material, F is the cross-section area of the rib, Js is the torque of the rib, S is the static moment of the n~o relative to the median surface of the casing,/1 is the bending moment of the rib out of the plane, J is the bending moment of the rib in the plane, w is the normal deflection, u and v are the displacements of the plate in the plane in the direction of the x andy axes, respectively,x andy are the coordinates in the plane, t is the time, E and E~ are Young's modulus of the materials of the plate and ribs, respectively, v is Poisson's ratio of the material of the plate, G~ is the shear modulus of the n~o material, h is the plate thickness, here and everywhere below the summation is from i = -(N + 1)/2 to i = (N + 1)/2, 5(...) is the Dirac delta-functiun, N is the number of n~os,and L~ and 2L2 are the lengths of the plate in the direction of the x and y axes, respectively. We will introduce slow and fast variables 11and 9, and represent the displacements in the form W=Wo-t-E4WI -b E5W2.-t-... (12) /4 ----U0 -I- E2Ul -I- E3U2 + .... U = U 0 -I- E2~l -I- E3U2+...
where functions of the zero approximation depend on the slow variables only. The variability with respect to time is assumed to be small, i.e. we are considering the lower part of the spectrum. For the parameters which appear in the boundary-value problem ( 1)-( 11) we take
-1, (~,, "/,, ~'0, ~1, k2, %, a~, J~)- e Substituting series (12) into system (1)-(3) and using the new expression for the derivative
= ~ +~-~Z
an, an
a~
by splitting with respect to e we obtain a system of equations which are integrated to give
w~ u? _4 + C l + C2~p+ %~02 + c4~03 w = -~(13)
_w2_2
ws
.2
u~--fv +%+%~, v~=-7--~
+CT+G~0
The condition of matching of adjacent parts of the plate (4)-(8) can be rewritten in the form
(wt, wlnl, ul, V l ) - = ( w l , WITll, UI, Vl) +
(14)
W?tp¢O -- Wl-q~ -~ ~,-IO~kWo~,q
(15)
wl-~0~ - w ~
= O~Wo~~ + Yu0~ + ~lw0~
ul-~ - u ~ = ~Uo~~ - ~,2Uo~ + YlWo~ v l~ - v l+n = ~l u 0 ~
- Boll~'2v 0=
(16) (17) (18)
Equations (16)-(18) are essentially the conditions of solvability of the equations of the first approximation, and yield at once the equations W1 + tzw0~ ~ + YUo~ + LlWo= = 0 W 2 + ~u0~ ~ - )~2Uor~ + y! Wo~,~, = 0
(19)
W3 + [tx~vo~:,~,~]- Bo~l~.:o.~= 0 System (19) corresponds to the constructive orthotropic theory. Its main feature is that it is of order ten with respect to ~, which raises the question of the boundary conditions. Conditions (9)-(10) become w0=wo n = u o = v o = 0
ifox 11=1/2
Wo=Wo~=Uo=Vo=O f o r : ~ = 0 , L The missing boundary conditions,are'obtained by averaging relations (11)
An asymptotic investigation of the dynamics of eccentrically reinforced plates
331
Vo~=0 for ~ = O, L
(20)
For symmetrically placed nq~s, system (19) splits up and the stiffness of the ribs in bending out of the plane is found to have no influence on the normal displacement w@ Moreover, for small values of that stiffness (¢zl - ~P, p > 1) the term in square brackets in Eqs (19) must be omitted and the boundary condition (20) dropped. In the original variables we have ['O4w 0 2 O2w0 04w0 P o h O 2 w o ] 2 L 4 2 Y ¢ 3 _ 4 Y ~ + 2 ¢ Y ~ 2 _ ( Y ] wl=-[.~"ff~ -+ ~ + ~y 4 + D ~t 2 b ~, b J ~,b J •
y
3]
Y
The expression for ~'1 is the same as ul with u0 replaced by v0. We now construct the physical relations. By splitting the initial relations with respect to e and averaging we obtain
M~o) = D(xlo) + ,,~(o)~~ EcJ ~(o)_L EcS o(o) "'~2 X T
b
"~I 7-'~--GI
M2 = D(~(2o)+ v×lO)), ~¢o) ~'*12 = D(I -,,~.¢o) *#~'12
EcF ~(o)l+ E~bS×I°)
Yl(0) = B(EI 0) + VE(0)~+2 / T
=
,4 °'
=--W0xx'
+ v lO)), r, o) --=t.,0c, o o 12O)
,4'
------W0yy'"12 =--WOxy
El 0) =UOx , E(0) =U0y , El7 ) =UOy+OOx Here xl = --w•, ~: ffi --wyy,xm ffi - w ~ ~1 = ux, ~2 = vy, e12 = uy + v , M1, M2, M12are, respectively, the bending moment and the torque, Xx, x2 are the curvatures, xm is the torsion, T1, T2 (~1, e2) are, respectively, the forces (deformations) in thex andy directions, Tm is the shear force, and ~12 is the shear strain. Thus, the method of averasing can be used to obtain both sequential averaged relations and corrections for discreteness in analytic form. This research was supported in part by the International Soros Program of Educational Support in the Exact Sciences CISSEP, SPU061002.
REFERENCES 1. AMIRO, I. Ya., ZARUTSKII, V. A. and POLYAKOV,R S., C y ~ l R/bbed Shells. Naukova Dumka; Kiev, 1973. 2. AMIRO, I. Ya. and ZARUTSKII, V. A., The Theory of Ribbed Shells. Naukova Dumka, Kiev, 1980. 3. AMIRO, I. Ya. and ZARUTSKII, V. A., The statics, dynamics and stability of fibbed shells. In Advances in Science and Technology. See. The Mechanics of a Deformable Solid., 1990, VoL 21, pp. 132-191. VINITI, Moscow. 4. KARMISHIN, A. V., LYASKOVETS, V. A., MYACHENKOV, V. I. and FROLOV, A. N., The Statics and Dynamics of Thinwalled Shell Structun'.s. M~qhinostroyeniye, Moscow, 1975. 5. ANDRIANOV, I. V., LESNICHAYA, V. A., and MANEVICH, L. I., The Method of Averasing in the Statics and Dynamics of
R/bbed Shells. Nanka, Moscow, 1985. 6. OBRAZTSOV, I. E, NERUBAILO, B. V. and ANDRIANOV, I. V., Asymptotic Methods in the Structural Mechanics of Thinwalled Structures. M.'tshinostroyoeniye, Moscow, 1991. 7. DUVAU'I~ G., Analyse fonctionnelle et m6canique des milieux continus. Application ~i l'6tude des materiaux composites 61astiques b structure periodique---homog6n6isation. In Theoretical and Applied Mechanics. Prec. of 14th IUTAM Congress, Delft, The Netherlands, 30 August-4 September 1976. North-Holland, Amsterdam, 1977. 8. KAI2kMKAROV, A. L, KUDRYAVTSEV, B. A. and PARTON, V. Z, An asymptotic method of averaging in the mechanics of composites of reg.~lar structure. In Advances in Science and Technology. Set. The Mechanics of a Deformable Solid. 1987, Vol. 19, pp. 78-147, VIN1TI, Moscow. 9. LEWINSKI, T., Homogenizing stiff~esses of plates with periodic structure. Int. J. Solids Structures 1992, 29, 309-326. Translated by R.L.
Pergamon PH: S0021-8928(97)00041.5
AN ASYMPTOTIC INVESTIGATION OF THE DYNAMICS OF ECCENTRICALLY REINFORCED PLATES? I. V. A N D R I A N O V
a n d V. I. P I S K U N O V
Dnepropetrovsk (Rece/ved 9 June 1995) With the aim of extending earlier results [1-9] obtained by the method of avere~ng [5-9], the equations of the vibrations of a plate reinforced by elm;tic n"es mounted e~ontrically with respect to its median surface are investigated. A system of recun~nce equations is derived, o:[ which the first approximation corresponds to the construedve orthotropie theory. It is shown that even this approximation reveals the singularities of asymmetricallydistn~mutedn'bs. Corrections for discretely situated ribs are obtained. © 1997 Elsevier Science Ltd. All rights reserved.
In dimensionless variables, the initial relations describing free vibrations of an eccentrically reinforced plate have the form
I++2
°34
a "~
(1)
=o
BotU~ + un,n, + (VBo, + l ) v gn, = u ~
(2)
Bo,v ~ +v n,~, + (VBo, + 1)u~n, =v
(3)
(w, w,1', u, v ) - = ( w ,
0)
w,1,, u, v )+
(5)
W-lTllql -- Wvhqt = EO~kw~lI
W,l,n,nL -
_
%, - %, = [~u~ - Y2u,, +
vnt -
(6)
n,n,n,= £Otk wv,~,[~,+ Yuv,~,~,+ ~lWv¢
W +
klw~~
(7)
n, = c q v ~ - B o l k z v ~
(s)
_ U +
w=wnl = u = v = 0 w=w~=u=v
=0
11,=~
for
for ~ = 0 ,
u¢Y~a(q)-i)=O
(9) (lO)
L=LI/(2L2)
for ~ = O ,
(11)
L
Here
, ( .o
.Bo. ~ ,
Gogg
_ Ec___SS
;k.2 pcF
EcF
E¢S
B Bol---~,
Eh 8=1_v2
kl = 2L2PcFB° , D = Bh2
DPoh E~J
E~Jj
12
Eh
2L~poh' ~'' = 4-~o~' '~' = 8-b-4' '~= Ob' B o - - -2 ( l + v ) (...)(-) = l i r a ( . . . ) , k = 0 + l y-.~/~_) +0
a2
..... + ( N + I ) / 2
a2
a2
,~ =,~a ~, a' = a--~-+ a-7. (...)~,. _=a--~y(... ) ?Prik£ Mat MeldL 3/hi. 61, No. 2, pp. 341-344, 1997. 329
330
I.V. Andrianov and V. I. Piskunov
Po is the density of the plate material per unit area, Pc is the density per unit length of the n~o material, F is the cross-section area of the rib, Js is the torque of the rib, S is the static moment of the n~o relative to the median surface of the casing,/1 is the bending moment of the rib out of the plane, J is the bending moment of the rib in the plane, w is the normal deflection, u and v are the displacements of the plate in the plane in the direction of the x andy axes, respectively,x andy are the coordinates in the plane, t is the time, E and E~ are Young's modulus of the materials of the plate and ribs, respectively, v is Poisson's ratio of the material of the plate, G~ is the shear modulus of the n~o material, h is the plate thickness, here and everywhere below the summation is from i = -(N + 1)/2 to i = (N + 1)/2, 5(...) is the Dirac delta-functiun, N is the number of n~os,and L~ and 2L2 are the lengths of the plate in the direction of the x and y axes, respectively. We will introduce slow and fast variables 11and 9, and represent the displacements in the form W=Wo-t-E4WI -b E5W2.-t-... (12) /4 ----U0 -I- E2Ul -I- E3U2 + .... U = U 0 -I- E2~l -I- E3U2+...
where functions of the zero approximation depend on the slow variables only. The variability with respect to time is assumed to be small, i.e. we are considering the lower part of the spectrum. For the parameters which appear in the boundary-value problem ( 1)-( 11) we take
-1, (~,, "/,, ~'0, ~1, k2, %, a~, J~)- e Substituting series (12) into system (1)-(3) and using the new expression for the derivative
= ~ +~-~Z
an, an
a~
by splitting with respect to e we obtain a system of equations which are integrated to give
w~ u? _4 + C l + C2~p+ %~02 + c4~03 w = -~(13)
_w2_2
ws
.2
u~--fv +%+%~, v~=-7--~
+CT+G~0
The condition of matching of adjacent parts of the plate (4)-(8) can be rewritten in the form
(wt, wlnl, ul, V l ) - = ( w l , WITll, UI, Vl) +
(14)
W?tp¢O -- Wl-q~ -~ ~,-IO~kWo~,q
(15)
wl-~0~ - w ~
= O~Wo~~ + Yu0~ + ~lw0~
ul-~ - u ~ = ~Uo~~ - ~,2Uo~ + YlWo~ v l~ - v l+n = ~l u 0 ~
- Boll~'2v 0=
(16) (17) (18)
Equations (16)-(18) are essentially the conditions of solvability of the equations of the first approximation, and yield at once the equations W1 + tzw0~ ~ + YUo~ + LlWo= = 0 W 2 + ~u0~ ~ - )~2Uor~ + y! Wo~,~, = 0
(19)
W3 + [tx~vo~:,~,~]- Bo~l~.:o.~= 0 System (19) corresponds to the constructive orthotropic theory. Its main feature is that it is of order ten with respect to ~, which raises the question of the boundary conditions. Conditions (9)-(10) become w0=wo n = u o = v o = 0
ifox 11=1/2
Wo=Wo~=Uo=Vo=O f o r : ~ = 0 , L The missing boundary conditions,are'obtained by averaging relations (11)
An asymptotic investigation of the dynamics of eccentrically reinforced plates
331
Vo~=0 for ~ = O, L
(20)
For symmetrically placed nq~s, system (19) splits up and the stiffness of the ribs in bending out of the plane is found to have no influence on the normal displacement w@ Moreover, for small values of that stiffness (¢zl - ~P, p > 1) the term in square brackets in Eqs (19) must be omitted and the boundary condition (20) dropped. In the original variables we have ['O4w 0 2 O2w0 04w0 P o h O 2 w o ] 2 L 4 2 Y ¢ 3 _ 4 Y ~ + 2 ¢ Y ~ 2 _ ( Y ] wl=-[.~"ff~ -+ ~ + ~y 4 + D ~t 2 b ~, b J ~,b J •
y
3]
Y
The expression for ~'1 is the same as ul with u0 replaced by v0. We now construct the physical relations. By splitting the initial relations with respect to e and averaging we obtain
M~o) = D(xlo) + ,,~(o)~~ EcJ ~(o)_L EcS o(o) "'~2 X T
b
"~I 7-'~--GI
M2 = D(~(2o)+ v×lO)), ~¢o) ~'*12 = D(I -,,~.¢o) *#~'12
EcF ~(o)l+ E~bS×I°)
Yl(0) = B(EI 0) + VE(0)~+2 / T
=
,4 °'
=--W0xx'
+ v lO)), r, o) --=t.,0c, o o 12O)
,4'
------W0yy'"12 =--WOxy
El 0) =UOx , E(0) =U0y , El7 ) =UOy+OOx Here xl = --w•, ~: ffi --wyy,xm ffi - w ~ ~1 = ux, ~2 = vy, e12 = uy + v , M1, M2, M12are, respectively, the bending moment and the torque, Xx, x2 are the curvatures, xm is the torsion, T1, T2 (~1, e2) are, respectively, the forces (deformations) in thex andy directions, Tm is the shear force, and ~12 is the shear strain. Thus, the method of averasing can be used to obtain both sequential averaged relations and corrections for discreteness in analytic form. This research was supported in part by the International Soros Program of Educational Support in the Exact Sciences CISSEP, SPU061002.
REFERENCES 1. AMIRO, I. Ya., ZARUTSKII, V. A. and POLYAKOV,R S., C y ~ l R/bbed Shells. Naukova Dumka; Kiev, 1973. 2. AMIRO, I. Ya. and ZARUTSKII, V. A., The Theory of Ribbed Shells. Naukova Dumka, Kiev, 1980. 3. AMIRO, I. Ya. and ZARUTSKII, V. A., The statics, dynamics and stability of fibbed shells. In Advances in Science and Technology. See. The Mechanics of a Deformable Solid., 1990, VoL 21, pp. 132-191. VINITI, Moscow. 4. KARMISHIN, A. V., LYASKOVETS, V. A., MYACHENKOV, V. I. and FROLOV, A. N., The Statics and Dynamics of Thinwalled Shell Structun'.s. M~qhinostroyeniye, Moscow, 1975. 5. ANDRIANOV, I. V., LESNICHAYA, V. A., and MANEVICH, L. I., The Method of Averasing in the Statics and Dynamics of
R/bbed Shells. Nanka, Moscow, 1985. 6. OBRAZTSOV, I. E, NERUBAILO, B. V. and ANDRIANOV, I. V., Asymptotic Methods in the Structural Mechanics of Thinwalled Structures. M.'tshinostroyoeniye, Moscow, 1991. 7. DUVAU'I~ G., Analyse fonctionnelle et m6canique des milieux continus. Application ~i l'6tude des materiaux composites 61astiques b structure periodique---homog6n6isation. In Theoretical and Applied Mechanics. Prec. of 14th IUTAM Congress, Delft, The Netherlands, 30 August-4 September 1976. North-Holland, Amsterdam, 1977. 8. KAI2kMKAROV, A. L, KUDRYAVTSEV, B. A. and PARTON, V. Z, An asymptotic method of averaging in the mechanics of composites of reg.~lar structure. In Advances in Science and Technology. Set. The Mechanics of a Deformable Solid. 1987, Vol. 19, pp. 78-147, VIN1TI, Moscow. 9. LEWINSKI, T., Homogenizing stiff~esses of plates with periodic structure. Int. J. Solids Structures 1992, 29, 309-326. Translated by R.L.