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GEOMETRICALLY NON-LINEAR ANALYSIS OF A THICK ANNULAR PLATE WITH ELASTICALLY CONSTRAINED EDGE USING GALERKIN'S METHOD
Journal of Sound and
LETTERS TO THE EDITOR GEOMETRICALLY NON-LINEAR ANALYSIS OF A THICK ANNULAR PLATE WITH ELASTICALLY CONSTRAINED EDGE USING GALERKIN'S METHOD P. C. DUMIR AND G. P. DUBE Applied Mechanics Department, I.I.¹. Delhi, New Delhi 110016, India. E-mail: [email protected] (Received 25 January 2001)
1. INTRODUCTION
Raju and Rao [1] and Reddy and Huang [2] have presented axisymmetric non-linear vibration of moderately thick circular and annular plates using the "nite-element method. Galerkin solutions using one-term spatial mode shape for de#ection, are presented by Sathyamoorthy [3, 4] for non-linear vibration of circular thick plates, employing "rst order shear deformation theory and Berger's approximation. Dumir and Shingal [5, 6] have presented geometrically non-linear static, transient and postbuckling analysis of thick annular plates, employing orthogonal point collocation method. Neetha et al. [7] have developed postbuckling analysis of a moderately thick elastic annular plate using the "nite-element method. This work presents an approximate closed-form analytical solution of the geometrically non-linear, axisymmetric, moderately large de#ection response of a polar orthotropic, moderately thick, annular plate with a free hole and elastically constrained outer edge. Neglecting the rotational and in-plane inertia, the geometrically non-linear "rst order shear deformation theory (FSDT) is formulated in terms of de#ection w, rotation t of the normal to the midplane and the stress function U. The rotation t is approximated by a one-term mode shape and the moment equation of equilibrium is solved exactly for w satisfying all out-of-plane boundary conditions. The compatibility equation is solved exactly for U satisfying the in-plane boundary conditions. The Galerkin's method is applied to the equation of motion to obtain Du$ng-type equation for the de#ection. The e!ect of the rotational and in-plane sti!nesses of the support and the thickness and orthotropic parameters on the buckling, postbuckling, static, transient and non-linear vibration response of orthotropic thick plates is studied.
2. GOVERNING EQUATIONS OF FSDT
Consider an axisymmetric deformation of an annulus of outer radius a, inner radius b, thickness h and density c, with mid-plane at z"0. For a plate with elastically restrained outer edge, with rotational and inplane sti!nesses kN , kN , subjected to applied in-plane radial @ G force resultant NM and uniformly distributed transverse load q, the boundary conditions at r"a are wN "0,
M "!kN tN , P @
0022-460X/01/380556#10 $35.00/0
N "UM /r"NM !kN uN , P G
(1)
2001 Academic Press
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LETTERS TO THE EDITOR
where wN , uN and tN are the transverse and radial displacements of the mid-plane, rotation of its normal and UM is the stress function. The simply supported (S), clamped (C) and movable (M), immovable (I) boundary conditions correspond to kN "0, R and kN "0, @ G R respectively. The variables are non-dimensionalized as o"r/a,
w"wN /h,
q"t(h/a)(E /c) , s"a/h,
u"uN a/h,
A* "AM * E h, ?@ ?@ m"b/a,
t"atM /h, A "AM /E h,
N"NM a/E h,
k "kN a/E h, G G
U"aUM /E h, D "DM /E h, ?@ ?@
Q"qa/E h,
k "kN a/E h, @ @
(2)
where E is a speci"c Young's modulus, (AM , DM )"(h, h/12)QM , AM *"AM \ and AM "k hG , XP with [QM , QM , QM ]"[E , l E , E ]/(1!l l ). E are Young's moduli, l are the P PF F F PF FP G GH Poisson ratios and G are shear moduli. The shear correction factor k is taken as 5/6. GH The dimensionless forms of the governing equations of the geometrically non-linear FSDT [8] are A* (oU #U )!A* U/o"!w /2,
(3)
D (t #t /o)!D t/o!sA (t#w )"0,
(4)
[sA o(t#w )#Uw ] /o#Q!wK "0,
(5)
t(m)#w (m)"0, D t (m)#D t(m)/m"0, D t (1)#D t(1)"!k t(1), @
(6)
w(1)"0, U(m)"0, U(1)"N!k [A* U(1)#A* U (1)], G
(7)
where ( ) "( ) , ( )"( ) . M O 3. METHOD OF SOLUTION
The approximate analytical solution is based on assumed spatial mode shapes for t, w having the same form as in the linear solution for static load Q. The rotation t is approximated as t"g(q) t oLG G G
(8)
with n "3, 1,!p, p; p"(D /D ) "(A* /A* ) , t "1. t , t and t are obtained G from equations (6) using equation (4): (9D !D )mt #(D !D )t /m"0, G (m)t #G (m)t #G (m)t "!G (m), \N N [G (1)#k ]t #[G (1)#k ]t #[G (1)#k ]t "!k !G (1), @ @ \N @ @ N
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with G (o)"(mD #D )oK\. Equation (4) is solved accurately, subject to condition K (7) , in order to yield w"g(q)
w oKG#w ln o , w "g(q) d oLG G G G G
(10)
with m "0, 4, 2, 1!p, 1#p, w "!t /4, w "!t /2#(9D !D )t /2sA , G w "t /(p!1), w "!1/(p#1), w "(D !D )t /sA , w "!w !w !w !w ,
d "m w , i"1,2, 4, d "w , n "!1. (11) G G> G>
Equation (3) is solved accurately, subject to conditions (7) and (7) , in order to yield the stress function U as U"g(q)
* ddI oLG>LH># (a #c ln o)oJI #N b oJI I I G H LG>LH> I K G H K
(12)
with l "p, l "!p and [a , a , b , b ]"[(x x !x x ), (x x !x x ), x , !x ]/(x x !x x ), [c , c ]"[!d , d ]d /2pA* ,
x "1#k (A* #pA* ), x "1#k (A* !pA* ), G G x "mN,
x "m\N,
* x " [1#k +A* #(n #n #1)A* ,]d d I #k A* (c #c ), G G H G H LG>LH> G G H * x " dd I mLG>LH># c mJK ln m, I "!1/2(qA* !A* ). O G H LG>LH> K G H K
(13)
The superscript * in the double sum denotes that the terms for (i, j )"(3, 5), (4, 5), (5, 3), (5, 4) are excluded. Substituting t, w, U from equations (8), (10) and (12) into equation (5), applying the Galerkin procedure of multiplying by ow and integrating from o"m to o"1, yields on integration by parts, an equation for g: B gK #(B #B N)g#B g"B Q,
(14)
where B "! JK w ![¸M !JK !mJ ln m]w /2, KG> G G B "(9D !D )t J #(D !D )t J , B " d b J , B "!JK , G K LG>JK G K
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* B " d (a J #c ¸M )# d d d I J , I K LI>JK K LI>JK G H I LG>LH> LG>LH>LI> I K G H I d oLGoOdo" d H , with H " oK do, G G LG>O K K G K G
J" O
H "(1!mK>)/(m#1) K JK "( J !mOJ )/q, O O
for mO!1 and H "!ln m, \ d oLGoO ln o do" d ¸ , ¸ " oK ln o do, G LG>O G K K G K G
¸M " O
¸ "![1#+(m#1) ln m!1,mK>]/(m#1) for mO!1 and ¸ "!(ln m)/2. K \ (15) The de#ection at the inner edge is given by w(m, q)"g (q)"B g(q) with B " w mKG#w ln m. The governing equation for de#ection at the inner edge is G G obtained from equation (14) as A gK #(A #A N)g #A g"A Q,
(16)
where A "B /B , A "B /B , A "B /B , A "B /(B ), A "B . The buckling load N and the postbuckling response is obtained by setting gK "0, Q"0 AP in equation (16): n"N/N "1#e g , AP
N "!A /A , e "A /A , AP
(17)
with e being the postbuckling parameter. The static response under uniformly distributed load Q is given by Q"a (1!n)g #a g"a g [1!n#e g ],
a "A /A ,
a "A /A .
(18)
The maximum de#ection g under step load Q is obtained by integrating equation (16): K?V Q "a (1!n)g /2#a g /4"a [1!n#e g /2]g /2. K?V K?V K?V K?V
(19)
The free non-linear vibrations are governed by gK #u (g #eg )"0,
u "u* [ca/E h] "[(1!n)A /A ] , e"e /(1!n), (20)
where u and u* are the dimensionless and dimensional fundamental linear frequencies. The ratio ¹ of the non-linear period ¹ for amplitude C to the linear period ¹ can be ! expressed in terms of the complete elliptic integral K of the "rst kind: ¹ "¹/¹ "2K(e)/n(1#eC) , e"[eC/2(1#eC)] . !
(21)
560
TABLE 1 Comparison of postbuckling response (b/a"0)4) !N CM
SM
Present
Collocation
Present
Collocation
s
g "0
g "1
g "2
g "0
g "1
g "2
g "0
g "1
g "2
g "0
g "1
g "2
1
100 10 6 100 10 6 100 10 6 100 10 6
1)803 1)773 1)706 2)961 2)852 2)643 7)413 6)675 5)549 4)388 4)198 3)737
2)020 1)980 1)896 3)538 3)393 3)130 8)980 8)105 6)803 4)607 4)391 3)898
2)672 2)599 2)464 5)268 5)015 4)593 13)679 12)392 10)564 5)285 4)970 4)380
1)690 1)632 1)536 2)822 2)675 2)444 7)315 6)498 5)375 4)120 3)774 3)272
1)941 1)879 1)776 3)485 3)318 3)052 8)957 8)026 6)716 4)375 4)022 3)508
2)653 2)572 2)436 5)267 5)003 4)575 13)559 12)104 9)902 5)125 4)741 4)176
0)250 0)249 0)246 0)741 0)728 0)705 2)386 2)271 2)089 0)230 0)229 0)226
0)356 0)355 0)351 1)007 0)993 0)968 3)096 2)979 2)795 0)337 0)336 0)333
0)674 0)671 0)666 1)807 1)788 1)758 5)223 5)103 4)915 0)660 0)658 0)653
0)248 0)246 0)243 0)741 0)728 0)705 2)348 2)227 2)038 0)230 0)228 0)226
0)353 0)351 0)347 1)001 0)986 0)959 3)027 2)891 2)675 0)337 0)336 0)333
0)649 0)644 0)635 1)729 1)696 1)640 5)070 4)816 4)402 0)652 0)647 0)638
100 10 6
2)053 1)966 1)821
2)193 2)103 1)950
k "1, k "0 @ G 2)627 2)012 2)515 1)918 2)335 1)769
2)169 2)072 1)918
2)624 2)515 2)338
4)328 4)134 3)839
5)052 4)865 4)557
k "k "1 @ G 7)225 4)242 7)026 4)044 6)710 3)729
4)980 4)783 4)470
7)191 6)994 6)686
3 10 1/3
1/3
LETTERS TO THE EDITOR
b
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4. RESULTS AND CONCLUSIONS
The results are non-dimensionalized with E taken to be equal to the smaller in-plane Young's modulus. Plates are analyzed with b"E /E "1, 3, 10, 1/3, having the greater F P in-plane Poisson ratio of 0)25 and G /E "0)4. The buckling and postbuckling loads of PX thick polar orthotropic annular plate with b/a"0)4 are compared in Table 1 with those obtained using orthogonal point collocation method of reference [6]. All the present results for the isotropic case are obtained using b"1)01. In most cases, the error is less than 5% for orthotropic plates and it generally increases with s and decreases with b. The buckling loads of CM plates are less accurate, but the postbuckling loads for g "2 are quite accurate. The non-linear vibration response of reference [1] using FEM, for isotropic thick circular plate, presented in Table 2, is in excellent agreement with the present Galerkin solution for a slightly orthotropic plate with a small hole with b"1)01, b/a"0)01. The maximum de#ection of plates with b/a"0)5 under step load Q "15 is compared in Table 3 with the collocation solution of reference [5]. There is very good agreement among these solutions with the maximum di!erence being 5%. The corresponding static response, not reported here for brevity, is also in good agreement. The response parameters of clamped and simply supported thick plates with b/a"0)3 are given in Table 4. The critical load N is for the movable inplane condition. The non-linear AP parameter for the immovable and movable inplane conditions is listed as e (I) and e (M) respectively. It is observed that the non-linearity parameter e is much more for the simply supported plates than the clamped ones. It is also much more for the immovable in-plane condition that the movable one. The in#uence of the thickness parameter s on e increases with the modular ratio b. The e!ect of s on a N , u is much more for the clamped plates AP than the simply supported ones and this e!ect increases with b. The dependence of the non-linear response parameter e , for plate with b/a"0)3, b"3 on the in-plane support sti!ness k , for the clamped (k "R) and the simply supported G @ (k "0) cases, is shown in Figure 1(a). The dependence of e for these plates on the @ rotational support sti!ness k , for the immovable (k "R) inplane condition, is also shown @ G in Figure 1(a). The rate of increase of e with k is predominant in the range of k "0)1}10. G G The rate of decrease of e with k is predominant in the range of k "0)2}2. The dependence @ @ of e for plates with b/a"0)3, b"3 on the rotational support sti!ness k , for the movable @ (k "0) in-plane condition and for the case of k "k , are presented in Figure 1(b). The G G @ e!ect of the rotational sti!ness k of movable support on N , a , u for these plates is shown @ AP in Figure 2. The rate of change of these response parameters is signi"cant in the intermediate range of k . @
TABLE 2 Comparison of non-linear vibration response (l "0)3) F SI Present s
u
1000 10 5
1)490 1)482 1)457
CI
Reference [1] (FEM)
¹ /¹
u
0)6517 0)6494 0)6424
1)494 1)481 1)446
Present
¹ /¹
u
0)6682 0)6670 0)6634
3)119 3)050 2)863
Reference [1] (FEM)
¹ /¹
u
0)8612 0)8598 0)8532
3)091 3)011 2)806
¹ /¹ 0)8607 0)8591 0)8533
562
TABLE 3 Maximum de-ection at the hole under step load Q "15 (b/a"0)5)
CM
SM
k "k "1 @ G
b"10
b"10
b"1/3
CI b"1
b"3
b"10
b"1/3
s
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
100 10 6
1)429 1)482 1)567
1)423 1)496 1)530
1)115 1)164 1)242
1)119 1)153 1)171
0)714 0)777 0)874
0)729 0)774 0)850
0)625 0)710 0)855
0)626 0)730 0)855
0)740 0)815 0)941
0)755 0)816 0)899
1)569 1)596 1)640
1)560 1)594 1)620
1)537 1)589 1)673
1)544 1)570 1)684
LETTERS TO THE EDITOR
w K?V
TABLE 4 Response parameters of annular plates (b/a"0)3) Clamped
Simply supported
s
a
u
!N (M) AP
e (I)
e (M)
a
u
!N (M) AP
e (I)
e (M)
1
100 10 6
6)947 6)707 6)316
3)478 3)413 3)302
1)480 1)459 1)412
0)3898 0)3904 0)3962
0)1506 0)1462 0)1403
1)184 1)177 1)165
1)398 1)393 1)384
0)279 0)277 0)274
1)938 1)956 1)987
0)3788 0)3803 0)3829
3
100 10 6
11)003 10)414 9)499
4)251 4)127 3)925
2)824 2)713 2)509
0)4131 0)4226 0)4472
0)1969 0)1939 0)1919
3)242 3)188 3)097
2)234 2)213 2)178
0)792 0)776 0)750
1)414 1)447 1)508
0)3207 0)3261 0)3359
10
100 10 6
22)222 19)941 16)837
5)743 5)433 4)977
7)509 6)746 5)609
0)3310 0)3554 0)4101
0)2006 0)2057 0)2195
9)552 9)104 8)398
3)656 3)568 3)425
2)441 2)324 2)139
0)968 1)018 1)107
0)2827 0)2965 0)3211
1/3
100 10 6
16)586 15)281 13)390
5)398 5)165 4)795
3)199 3)107 2)862
0)2004 0)2004 0)2121
0)0727 0)0675 0)0626
1)122 1)115 1)104
1)370 1)365 1)357
0)252 0)250 0)247
2)349 2)368 2)403
0)4462 0)4475 0)4498
LETTERS TO THE EDITOR
b
563
564
LETTERS TO THE EDITOR
Figure 1. Variation of non-linearity parameter e for plates with b/a"0)3, b"3, (a) with k , k : (1) k "R, G @ @ (2) k "0, (3) k "R and (b) with k : (1) k "0, (2) k "k ; (**), S"100; (} } }), S"6. @ G @ G G @
It is concluded that the present uni"ed simple approximate analytical solution for non-linear static, vibratory and transient response of thick orthotropic annular plate under axisymmetric transverse and in-plane loads yields accurate results. REFERENCES 1. K. KANAKA RAJU and G. VENKATESWARA RAO 1976 Journal of Sound and
LETTERS TO THE EDITOR
565
Figure 2. The e!ect of rotational sti!ness k of a movable support on N , a , u . (22), S"100; (} } }), S"6. @ AP
4. M. SATHYAMOORTHY 1996 Journal of Sound and
LETTERS TO THE EDITOR GEOMETRICALLY NON-LINEAR ANALYSIS OF A THICK ANNULAR PLATE WITH ELASTICALLY CONSTRAINED EDGE USING GALERKIN'S METHOD P. C. DUMIR AND G. P. DUBE Applied Mechanics Department, I.I.¹. Delhi, New Delhi 110016, India. E-mail: [email protected] (Received 25 January 2001)
1. INTRODUCTION
Raju and Rao [1] and Reddy and Huang [2] have presented axisymmetric non-linear vibration of moderately thick circular and annular plates using the "nite-element method. Galerkin solutions using one-term spatial mode shape for de#ection, are presented by Sathyamoorthy [3, 4] for non-linear vibration of circular thick plates, employing "rst order shear deformation theory and Berger's approximation. Dumir and Shingal [5, 6] have presented geometrically non-linear static, transient and postbuckling analysis of thick annular plates, employing orthogonal point collocation method. Neetha et al. [7] have developed postbuckling analysis of a moderately thick elastic annular plate using the "nite-element method. This work presents an approximate closed-form analytical solution of the geometrically non-linear, axisymmetric, moderately large de#ection response of a polar orthotropic, moderately thick, annular plate with a free hole and elastically constrained outer edge. Neglecting the rotational and in-plane inertia, the geometrically non-linear "rst order shear deformation theory (FSDT) is formulated in terms of de#ection w, rotation t of the normal to the midplane and the stress function U. The rotation t is approximated by a one-term mode shape and the moment equation of equilibrium is solved exactly for w satisfying all out-of-plane boundary conditions. The compatibility equation is solved exactly for U satisfying the in-plane boundary conditions. The Galerkin's method is applied to the equation of motion to obtain Du$ng-type equation for the de#ection. The e!ect of the rotational and in-plane sti!nesses of the support and the thickness and orthotropic parameters on the buckling, postbuckling, static, transient and non-linear vibration response of orthotropic thick plates is studied.
2. GOVERNING EQUATIONS OF FSDT
Consider an axisymmetric deformation of an annulus of outer radius a, inner radius b, thickness h and density c, with mid-plane at z"0. For a plate with elastically restrained outer edge, with rotational and inplane sti!nesses kN , kN , subjected to applied in-plane radial @ G force resultant NM and uniformly distributed transverse load q, the boundary conditions at r"a are wN "0,
M "!kN tN , P @
0022-460X/01/380556#10 $35.00/0
N "UM /r"NM !kN uN , P G
(1)
2001 Academic Press
557
LETTERS TO THE EDITOR
where wN , uN and tN are the transverse and radial displacements of the mid-plane, rotation of its normal and UM is the stress function. The simply supported (S), clamped (C) and movable (M), immovable (I) boundary conditions correspond to kN "0, R and kN "0, @ G R respectively. The variables are non-dimensionalized as o"r/a,
w"wN /h,
q"t(h/a)(E /c) , s"a/h,
u"uN a/h,
A* "AM * E h, ?@ ?@ m"b/a,
t"atM /h, A "AM /E h,
N"NM a/E h,
k "kN a/E h, G G
U"aUM /E h, D "DM /E h, ?@ ?@
Q"qa/E h,
k "kN a/E h, @ @
(2)
where E is a speci"c Young's modulus, (AM , DM )"(h, h/12)QM , AM *"AM \ and AM "k hG , XP with [QM , QM , QM ]"[E , l E , E ]/(1!l l ). E are Young's moduli, l are the P PF F F PF FP G GH Poisson ratios and G are shear moduli. The shear correction factor k is taken as 5/6. GH The dimensionless forms of the governing equations of the geometrically non-linear FSDT [8] are A* (oU #U )!A* U/o"!w /2,
(3)
D (t #t /o)!D t/o!sA (t#w )"0,
(4)
[sA o(t#w )#Uw ] /o#Q!wK "0,
(5)
t(m)#w (m)"0, D t (m)#D t(m)/m"0, D t (1)#D t(1)"!k t(1), @
(6)
w(1)"0, U(m)"0, U(1)"N!k [A* U(1)#A* U (1)], G
(7)
where ( ) "( ) , ( )"( ) . M O 3. METHOD OF SOLUTION
The approximate analytical solution is based on assumed spatial mode shapes for t, w having the same form as in the linear solution for static load Q. The rotation t is approximated as t"g(q) t oLG G G
(8)
with n "3, 1,!p, p; p"(D /D ) "(A* /A* ) , t "1. t , t and t are obtained G from equations (6) using equation (4): (9D !D )mt #(D !D )t /m"0, G (m)t #G (m)t #G (m)t "!G (m), \N N [G (1)#k ]t #[G (1)#k ]t #[G (1)#k ]t "!k !G (1), @ @ \N @ @ N
(9)
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with G (o)"(mD #D )oK\. Equation (4) is solved accurately, subject to condition K (7) , in order to yield w"g(q)
w oKG#w ln o , w "g(q) d oLG G G G G
(10)
with m "0, 4, 2, 1!p, 1#p, w "!t /4, w "!t /2#(9D !D )t /2sA , G w "t /(p!1), w "!1/(p#1), w "(D !D )t /sA , w "!w !w !w !w ,
d "m w , i"1,2, 4, d "w , n "!1. (11) G G> G>
Equation (3) is solved accurately, subject to conditions (7) and (7) , in order to yield the stress function U as U"g(q)
* ddI oLG>LH># (a #c ln o)oJI #N b oJI I I G H LG>LH> I K G H K
(12)
with l "p, l "!p and [a , a , b , b ]"[(x x !x x ), (x x !x x ), x , !x ]/(x x !x x ), [c , c ]"[!d , d ]d /2pA* ,
x "1#k (A* #pA* ), x "1#k (A* !pA* ), G G x "mN,
x "m\N,
* x " [1#k +A* #(n #n #1)A* ,]d d I #k A* (c #c ), G G H G H LG>LH> G G H * x " dd I mLG>LH># c mJK ln m, I "!1/2(qA* !A* ). O G H LG>LH> K G H K
(13)
The superscript * in the double sum denotes that the terms for (i, j )"(3, 5), (4, 5), (5, 3), (5, 4) are excluded. Substituting t, w, U from equations (8), (10) and (12) into equation (5), applying the Galerkin procedure of multiplying by ow and integrating from o"m to o"1, yields on integration by parts, an equation for g: B gK #(B #B N)g#B g"B Q,
(14)
where B "! JK w ![¸M !JK !mJ ln m]w /2, KG> G G B "(9D !D )t J #(D !D )t J , B " d b J , B "!JK , G K LG>JK G K
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LETTERS TO THE EDITOR
* B " d (a J #c ¸M )# d d d I J , I K LI>JK K LI>JK G H I LG>LH> LG>LH>LI> I K G H I d oLGoOdo" d H , with H " oK do, G G LG>O K K G K G
J" O
H "(1!mK>)/(m#1) K JK "( J !mOJ )/q, O O
for mO!1 and H "!ln m, \ d oLGoO ln o do" d ¸ , ¸ " oK ln o do, G LG>O G K K G K G
¸M " O
¸ "![1#+(m#1) ln m!1,mK>]/(m#1) for mO!1 and ¸ "!(ln m)/2. K \ (15) The de#ection at the inner edge is given by w(m, q)"g (q)"B g(q) with B " w mKG#w ln m. The governing equation for de#ection at the inner edge is G G obtained from equation (14) as A gK #(A #A N)g #A g"A Q,
(16)
where A "B /B , A "B /B , A "B /B , A "B /(B ), A "B . The buckling load N and the postbuckling response is obtained by setting gK "0, Q"0 AP in equation (16): n"N/N "1#e g , AP
N "!A /A , e "A /A , AP
(17)
with e being the postbuckling parameter. The static response under uniformly distributed load Q is given by Q"a (1!n)g #a g"a g [1!n#e g ],
a "A /A ,
a "A /A .
(18)
The maximum de#ection g under step load Q is obtained by integrating equation (16): K?V Q "a (1!n)g /2#a g /4"a [1!n#e g /2]g /2. K?V K?V K?V K?V
(19)
The free non-linear vibrations are governed by gK #u (g #eg )"0,
u "u* [ca/E h] "[(1!n)A /A ] , e"e /(1!n), (20)
where u and u* are the dimensionless and dimensional fundamental linear frequencies. The ratio ¹ of the non-linear period ¹ for amplitude C to the linear period ¹ can be ! expressed in terms of the complete elliptic integral K of the "rst kind: ¹ "¹/¹ "2K(e)/n(1#eC) , e"[eC/2(1#eC)] . !
(21)
560
TABLE 1 Comparison of postbuckling response (b/a"0)4) !N CM
SM
Present
Collocation
Present
Collocation
s
g "0
g "1
g "2
g "0
g "1
g "2
g "0
g "1
g "2
g "0
g "1
g "2
1
100 10 6 100 10 6 100 10 6 100 10 6
1)803 1)773 1)706 2)961 2)852 2)643 7)413 6)675 5)549 4)388 4)198 3)737
2)020 1)980 1)896 3)538 3)393 3)130 8)980 8)105 6)803 4)607 4)391 3)898
2)672 2)599 2)464 5)268 5)015 4)593 13)679 12)392 10)564 5)285 4)970 4)380
1)690 1)632 1)536 2)822 2)675 2)444 7)315 6)498 5)375 4)120 3)774 3)272
1)941 1)879 1)776 3)485 3)318 3)052 8)957 8)026 6)716 4)375 4)022 3)508
2)653 2)572 2)436 5)267 5)003 4)575 13)559 12)104 9)902 5)125 4)741 4)176
0)250 0)249 0)246 0)741 0)728 0)705 2)386 2)271 2)089 0)230 0)229 0)226
0)356 0)355 0)351 1)007 0)993 0)968 3)096 2)979 2)795 0)337 0)336 0)333
0)674 0)671 0)666 1)807 1)788 1)758 5)223 5)103 4)915 0)660 0)658 0)653
0)248 0)246 0)243 0)741 0)728 0)705 2)348 2)227 2)038 0)230 0)228 0)226
0)353 0)351 0)347 1)001 0)986 0)959 3)027 2)891 2)675 0)337 0)336 0)333
0)649 0)644 0)635 1)729 1)696 1)640 5)070 4)816 4)402 0)652 0)647 0)638
100 10 6
2)053 1)966 1)821
2)193 2)103 1)950
k "1, k "0 @ G 2)627 2)012 2)515 1)918 2)335 1)769
2)169 2)072 1)918
2)624 2)515 2)338
4)328 4)134 3)839
5)052 4)865 4)557
k "k "1 @ G 7)225 4)242 7)026 4)044 6)710 3)729
4)980 4)783 4)470
7)191 6)994 6)686
3 10 1/3
1/3
LETTERS TO THE EDITOR
b
561
LETTERS TO THE EDITOR
4. RESULTS AND CONCLUSIONS
The results are non-dimensionalized with E taken to be equal to the smaller in-plane Young's modulus. Plates are analyzed with b"E /E "1, 3, 10, 1/3, having the greater F P in-plane Poisson ratio of 0)25 and G /E "0)4. The buckling and postbuckling loads of PX thick polar orthotropic annular plate with b/a"0)4 are compared in Table 1 with those obtained using orthogonal point collocation method of reference [6]. All the present results for the isotropic case are obtained using b"1)01. In most cases, the error is less than 5% for orthotropic plates and it generally increases with s and decreases with b. The buckling loads of CM plates are less accurate, but the postbuckling loads for g "2 are quite accurate. The non-linear vibration response of reference [1] using FEM, for isotropic thick circular plate, presented in Table 2, is in excellent agreement with the present Galerkin solution for a slightly orthotropic plate with a small hole with b"1)01, b/a"0)01. The maximum de#ection of plates with b/a"0)5 under step load Q "15 is compared in Table 3 with the collocation solution of reference [5]. There is very good agreement among these solutions with the maximum di!erence being 5%. The corresponding static response, not reported here for brevity, is also in good agreement. The response parameters of clamped and simply supported thick plates with b/a"0)3 are given in Table 4. The critical load N is for the movable inplane condition. The non-linear AP parameter for the immovable and movable inplane conditions is listed as e (I) and e (M) respectively. It is observed that the non-linearity parameter e is much more for the simply supported plates than the clamped ones. It is also much more for the immovable in-plane condition that the movable one. The in#uence of the thickness parameter s on e increases with the modular ratio b. The e!ect of s on a N , u is much more for the clamped plates AP than the simply supported ones and this e!ect increases with b. The dependence of the non-linear response parameter e , for plate with b/a"0)3, b"3 on the in-plane support sti!ness k , for the clamped (k "R) and the simply supported G @ (k "0) cases, is shown in Figure 1(a). The dependence of e for these plates on the @ rotational support sti!ness k , for the immovable (k "R) inplane condition, is also shown @ G in Figure 1(a). The rate of increase of e with k is predominant in the range of k "0)1}10. G G The rate of decrease of e with k is predominant in the range of k "0)2}2. The dependence @ @ of e for plates with b/a"0)3, b"3 on the rotational support sti!ness k , for the movable @ (k "0) in-plane condition and for the case of k "k , are presented in Figure 1(b). The G G @ e!ect of the rotational sti!ness k of movable support on N , a , u for these plates is shown @ AP in Figure 2. The rate of change of these response parameters is signi"cant in the intermediate range of k . @
TABLE 2 Comparison of non-linear vibration response (l "0)3) F SI Present s
u
1000 10 5
1)490 1)482 1)457
CI
Reference [1] (FEM)
¹ /¹
u
0)6517 0)6494 0)6424
1)494 1)481 1)446
Present
¹ /¹
u
0)6682 0)6670 0)6634
3)119 3)050 2)863
Reference [1] (FEM)
¹ /¹
u
0)8612 0)8598 0)8532
3)091 3)011 2)806
¹ /¹ 0)8607 0)8591 0)8533
562
TABLE 3 Maximum de-ection at the hole under step load Q "15 (b/a"0)5)
CM
SM
k "k "1 @ G
b"10
b"10
b"1/3
CI b"1
b"3
b"10
b"1/3
s
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
Gal.
Coll.
100 10 6
1)429 1)482 1)567
1)423 1)496 1)530
1)115 1)164 1)242
1)119 1)153 1)171
0)714 0)777 0)874
0)729 0)774 0)850
0)625 0)710 0)855
0)626 0)730 0)855
0)740 0)815 0)941
0)755 0)816 0)899
1)569 1)596 1)640
1)560 1)594 1)620
1)537 1)589 1)673
1)544 1)570 1)684
LETTERS TO THE EDITOR
w K?V
TABLE 4 Response parameters of annular plates (b/a"0)3) Clamped
Simply supported
s
a
u
!N (M) AP
e (I)
e (M)
a
u
!N (M) AP
e (I)
e (M)
1
100 10 6
6)947 6)707 6)316
3)478 3)413 3)302
1)480 1)459 1)412
0)3898 0)3904 0)3962
0)1506 0)1462 0)1403
1)184 1)177 1)165
1)398 1)393 1)384
0)279 0)277 0)274
1)938 1)956 1)987
0)3788 0)3803 0)3829
3
100 10 6
11)003 10)414 9)499
4)251 4)127 3)925
2)824 2)713 2)509
0)4131 0)4226 0)4472
0)1969 0)1939 0)1919
3)242 3)188 3)097
2)234 2)213 2)178
0)792 0)776 0)750
1)414 1)447 1)508
0)3207 0)3261 0)3359
10
100 10 6
22)222 19)941 16)837
5)743 5)433 4)977
7)509 6)746 5)609
0)3310 0)3554 0)4101
0)2006 0)2057 0)2195
9)552 9)104 8)398
3)656 3)568 3)425
2)441 2)324 2)139
0)968 1)018 1)107
0)2827 0)2965 0)3211
1/3
100 10 6
16)586 15)281 13)390
5)398 5)165 4)795
3)199 3)107 2)862
0)2004 0)2004 0)2121
0)0727 0)0675 0)0626
1)122 1)115 1)104
1)370 1)365 1)357
0)252 0)250 0)247
2)349 2)368 2)403
0)4462 0)4475 0)4498
LETTERS TO THE EDITOR
b
563
564
LETTERS TO THE EDITOR
Figure 1. Variation of non-linearity parameter e for plates with b/a"0)3, b"3, (a) with k , k : (1) k "R, G @ @ (2) k "0, (3) k "R and (b) with k : (1) k "0, (2) k "k ; (**), S"100; (} } }), S"6. @ G @ G G @
It is concluded that the present uni"ed simple approximate analytical solution for non-linear static, vibratory and transient response of thick orthotropic annular plate under axisymmetric transverse and in-plane loads yields accurate results. REFERENCES 1. K. KANAKA RAJU and G. VENKATESWARA RAO 1976 Journal of Sound and
LETTERS TO THE EDITOR
565
Figure 2. The e!ect of rotational sti!ness k of a movable support on N , a , u . (22), S"100; (} } }), S"6. @ AP
4. M. SATHYAMOORTHY 1996 Journal of Sound and