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Buckling analysis of a laminated cylindrical shellunder torsion subjected to mixed boundary conditions
\ PERGAMON
International Journal of Solids and Structures 25 "0888# 2710Ð2724
Buckling analysis of a laminated cylindrical shell under torsion subjected to mixed boundary conditions Renjie Maoa\ \ C[!H[ Lub a
b
Department of En`ineerin` Mechanics\ Shan`hai Jiao Ton` University\ Shan`hai 199929\ P[R[ China R+D Senior En`ineer\ Mechanical Systems\ Dover Elevator Systems\ Inc[\ Horn Lake\ MS 27526\ U[S[A[ Received 17 October 0886^ in revised form 11 May 0887
Abstract A new e.cient method is developed in this paper for buckling analysis of a cross!ply laminated cylindrical shell under torsion subjected to mixed boundary conditions[ The transverse shear is taken into account by a _rst!order theory with a shear correction factor of 4:5[ The mixed boundary conditions include conditions in forces as well as conditions in displacements\ and these forces and displacements are selected as basic unknowns[ The other displacements and forces are expressed in terms of the basic unknowns by taking inverse of a matrix composed of operators[ The equations of buckled equilibrium in terms of the basic unknowns are solved with double trigonometric series which satisfy the mixed boundary conditions[ Com! parison of the obtained numerical results with those given in the literature based on completely clamped boundary conditions checks with the fact that the mixed boundary conditions yield appreciably lower buckling load and less circumferential wave number than the completely clamped boundary conditions[ The curves in the _gures show how the di}erence in buckling loads between the two kinds of boundary conditions varies when the length and thickness of the shell vary[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The buckling of cylindrical shells under torsion "torsional buckling# has been much less studied in contrast to the buckling under axial compression "compressive buckling#\ especially for anisotropic shells[ The main di.culty involved in the torsional buckling is in that _rstly the nonlinear terms in the equations of buckled equilibrium include partial derivatives both in axial direction and in circumferential direction\ so that the single!wave buckling mode\ which is useful in the analysis of compressive buckling\ does not work\ and secondly the boundary conditions in torsional force Txy and in torsional moment Mxy can hardly be satis_ed beforehand by the assumed buckling mode[ Simitses and Shaw "0874# and Hui and Du "0876# solve the di.culty for Donnell!type shallow
Corresponding author[ 9919Ð6572:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII ] S 9 9 1 9 Ð 6 5 7 2 " 8 7 # 9 9 0 6 7 Ð 3
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shells without transverse shear by introducing a stress function and assuming a multi!wave buckling mode such as truncated trigonometric series in u with the coe.cients being functions of x\ where x is the axial coordinate in length and u is the circumferential coordinate in angle[ Thus the equations of equilibrium and compatibility reduce to ordinary di}erential equations for the coe.cients of the series\ and the original boundary conditions yield boundary conditions for the solution of these ordinary di}erential equations[ In principle\ the above solution procedure may apply to non!shallow shells with transverse shear[ But in this case the number of the unknowns will be _ve and\ due to the use of non!shallow shell theory\ no simple stress function can be introduced to reduce that number[ The above solution procedure will then be much more involved[ To solve the problem\ Tabiei and Simitses "0883# express the equations of equilibrium in terms of the three displacements and two rotations and solve them with double trigonometric series which vanish at the boundaries[ Tabiei and Simitses| method is useful when all boundary conditions are in displacements and rotations\ i[e[\ the shell is completely clamped at its ends\ or when the shell is long enough so that the boundary conditions play little role[ However\ in the case of mixed boundary conditions\ some of the conditions are given as boundary forces[ They can hardly be satis_ed beforehand by the double trigonometric series for displacements and rotations assumed by Tabiei and Simitses "0883#[ That is why the torsional buckling of transversely shear deformable shells under mixed boundary conditions has not yet been studied much[ In many practical cases\ however\ the real physical boundaries may be described better by mixed boundary conditions than by completely clamped boundary conditions[ Therefore\ solution methods are needed to deal with mixed boundary conditions[ The present study undertakes this task to develop such a method[ In the present study\ the displacements "including rotations# and forces "including moments# appearing in the mixed boundary conditions are selected as basic unknowns and are expressed in terms of di}erent kinds of double trigonometric series which satisfy the mixed boundary conditions[ Then\ by taking inverse of a matrix composed of operators\ all other displacements and forces\ and hence the _ve equations of buckled equilibrium\ can be expressed in terms of the _ve basic unknowns[ Finally\ the application of the Galerkin procedure to the equations of buckled equilibrium leads to an eigenvalue problem[ The numerical results of examples show that the mixed boundary conditions lead to lower buckling load and less cir! cumferential wave number of the buckling mode than the completely clamped boundary conditions[
1[ Basic equations A thin cross!ply cylindrical shell subjected to torsion is under consideration[ In the _rst!order shear!deformation theory the displacement _eld of a cylindrical shell is ux u¦zc\ uy v¦z8\ uz w
"0#
where "x\ y\ z# are\ respectively\ the axial\ circumferential and normal coordinates in length\ "ux\ uy\ uz# are their corresponding displacement components at an arbitrary point of the shell\ and "u\ v\ w# are those components at the middle surface[ The linear strains at an arbitrary point of the shell can be derived from the displacement _eld\
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0
1u 1v w 1u 1v 1c 18 1c 18 ¦z \ o¹y ¦ ¦z \ g¹ xy ¦ ¦z ¦ 1x 1x 1y R 1y 1y 1x 1y 1x
o¹x
1
1w 1w y ¦c\ g¹ yz − ¦8 1x 1y R
g¹ xz
2712
"1#
These strains coincide with the general _rst!order shear!deformation kinematic relations of Tabiei and Simitses "0883# for linear case\ if the thin shell assumption z:R ð 0 is taken into account[ The de_nitions of the stress resultants are as usual
g
g
g g
g
Tx sx dz\ Ty sy dz\ Mx sx z dz\ My sy z dz
g
Txy Tyx txy dz\ Mxy Myx txy z dz
g
g
Qx txz dz\ Qy tyz dz
"2#
where
g g
h:1
\ h is the thickness of the shell
−h:1
By introducing vectors "s# "Tx \ Ty \ Txy \ Mx \ My \ Mxy \ Qy \ Qx # T "o# "ox \ oy \ gxy \ kx \ ky \ kxy \ gyz \ gxz # T where ox
1u 1v w 1u 1v \ oy ¦ \ gxy ¦ 1x 1y R 1y 1x
kx
1c 18 1c 18 \ ky \ kxy ¦ \ gyz g¹ yz \ gxz g¹ xz 1x 1y 1y 1x
are the strains and curvature changes of the middle surface\ the constitutive equations for cross! ply laminates can be written in matrix form "s# ðCŁ"o#
"3#
where
&
ðAŁ
ðBŁ
ðCŁ ðBŁ
ðDŁ ðA Ł
'
&
A00 A01
\
' &
B00 B01
ðAŁ A01 A11
\
A55
ðBŁ B01 B11 B55
'
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R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
&
D00 D01
ðDŁ D01 D11 D55
'
\
ðA Ł
$
A33 A44
%
The equations of equilibrium developed by Stein "0875# are used in the present study[ For a thin shell\ after imposing the following three assumptions about the nonlinear terms] "a# The only load is the boundary torques\ so that all nonlinear terms not related to the torsional force Txy are neglected^ "b# All nonlinear terms not related to w or its derivatives are neglected^ "c# All nonlinear terms not in the equation of normal equilibrium are neglected^ these equations reduce to 1Tx 1Txy ¦ 9 1x 1y
"4a#
1Txy 1Ty Qy ¦ ¦ 9 1x 1y R
"4b#
11 w 1Qx 1Qy Ty ¦ − ¦1Txy 9 1x 1y R 1x 1y
"4c#
1Mx 1Mxy ¦ −Qx 9 1x 1y
"4d#
1Mxy 1My ¦ −Qy 9 1x 1y
"4e#
and the variationally consistent boundary conditions can be written in the form "Tx du#= 9L 9
"5a#
ð"Txy −S#dvŁ= 9L 9
"5b#
$0
Txy
L
1 %b
1w ¦Qx dw 1y
9
"5c#
9
"Mx dc#= 9L 9
"5d#
"Mxy d8#= 9L 9
"5e#
"Txy du#= 91p 9
"6a#
"Ty dv#= 91p 9
"6b#
$0
Txy
1p
1 %b
1w ¦Qy dw 1x
9
9
"6c#
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"Mxy dc#= 91p 9
"6d#
"My d8#= 91p 9
"6e#
The symbols = 9L and = 91p are de_ned as "Tx du#= 9L "Tx du# x L −"Tx du# x 9 \ and so on^ "Txy du#= 91p "Txy du# u 1p −"Txy du# u 9 \ and so on where L and R are\ respectively\ the length and radius of the shell\ u y:R and S is the torsional force per unit length of y produced by the boundary torques[ Conditions "6aÐe# can be satis_ed by the requirement that all displacements and forces appearing in these conditions be periodic in u with a period equal to 1p[ For conditions "5aÐe#\ the following mixed boundary conditions are assumed Tx 9\ Txy S\ w 9\ Mx Mxy 9 at x 9\ L
"7#
It is obvious that conditions "5aÐe# will be satis_ed if conditions "7# are satis_ed[ Equilibrium equations "4aÐe# and boundary conditions "5a#Ð"6e# coincide with those of Tabiei and Simitses "0883# for thin shells using the _rst!order shear!deformation kinematic relations[ The equations of Tabiei and Simitses "0883# is based on Donnell!type theory[ This suggests that the error caused by the preceding assumptions "a#Ð"c# can be expected not to be larger than that of Donnell!type theory[ More complete equations of equilibrium for a thin cylindrical shell are given by Flugge "0859#[ These equations correspond to a system of kinematic relations in which all nonlinear terms of u\ v and w are retained[
2[ Solution The _ve variables Tx\ Txy\ w\ Mx and Mxy appearing in the mixed boundary conditions "7# are selected as basic unknowns[ The equations of buckled equilibrium "4aÐe# can be transformed into a system of _ve equations in the basic unknowns by eliminating all other displacements and forces[ To do this\ the displacements and rotations u\ v\ c and 8 are expressed in terms of the basic unknowns by using eqn "3# to obtain F u J F Tx −A01 w:R J G G G G v f jMx −B01 w:Rf j ðGŁ J F J F Txy c G G G G Mxy j f 8j f where
"8#
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K 1 1 1 1L HA00 A01 B00 B01 H 1x 1y 1x 1y H H H 1 1 d 1H HB00 B01 D00 D01 H 1x 1y dx 1yH ðGŁ H H 1 1 1 1H A55 B55 B55 H HA55 1y 1x 1y 1x H H H 1 1 1 1H B55 D55 D55 H H B55 1y 1x 1y 1xl k
Let Gij represent the elements of ðGŁ\ Lij be the cofactor of the element Gji\ and L be the determinant of ðGŁ[ They are all operators[ The inverse of ðGŁ can be written as ðGŁ −0 L−0 ðLij Ł
"09#
Substitution of eqn "09# into eqn "8# gives F Tx −A01 w:R J FuJ G G G G Mx −B01 w:R f vf j j L J F ðLij Ł J F Txy c G G G G Mxy f j f8j
"00#
The stress resultants Ty\ My\ Qx and Qy can then be expressed in terms of the basic unknowns through the constitutive equations] Ty A01
Qy A33
1 1
"01a#
1u 1v w 1c 18 ¦B11 ¦ ¦D01 ¦D11 1x 1y R 1x 1y
"01b#
My B01 Qx A44
0 0 1
1u 1v w 1c 18 ¦A11 ¦ ¦B01 ¦B11 1x 1y R 1x 1y
0 0
1w ¦c 1x
1w v − ¦8 1y R
"01c#
1
"01d#
By application of the operator L to each side of eqns "4bÐe# and using eqns "01aÐd# and eqn "00# to eliminate Ty\ My\ Qx and Qy\ eqns "4aÐe# can be transformed into a system of _ve _nal equations in the _ve basic unknowns[ These _nal equations are too lengthy to be presented here[ The _ve _nal equations are solved with truncated double trigonometric series Tx SSF1sin am x"Tmn cos nu¦T Þmn sin nu#
"02a#
Txy S¦S"cos am−0 x−cos am¦0 x#"Smn cos nu¦S Þmn sin nu#
"02b#
w S sin am x"Wmn cos nu¦W Þ mn sin nu#
"02c#
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Mx S sin am x"Hmn cos nu¦H Þ mn sin nu#
"02d#
Mxy S"cos am−0 x−cos am¦0 x#"Fmn cos nu¦F Þmn sin nu#
"02e#
where\ and in the following\ the symbols M
M
S s s \ am m 0 n 0
mp L
Tmn\ [ [ [ \ Þ Fmn are to be determined[ are used for brevity in writing[ The constant coe.cients Tmn\ Þ Trigonometric series "02# satisfy the mixed boundary conditions "7#[ Before using the Galerkin procedure\ a closer inspection of the operators in eqns "00# is useful[ The operator L is a fourth!order homogeneous polynomial in 1:1x and 1:1y[ Each term of the polynomial includes one of the operators 13:1x3\ 13:"1x1 1y1# and 13:1y3[ Therefore\ when the operator L is applied to a product of trigonometric functions such as sin amx cos nu\ a very simple result can be obtained L"sin am x cos nu# ðLŁmn sin am x cos nu
"03#
where the symbol ðLŁmn is a number denoting the determinant of a matrix obtained by replacing 1:1x and 1:1y with am and bn\ respectively\ in the operator L\ where bn n:R[ Each of the operators Lij "i\ j 0\ 1\ 2\ 3# is a third!order polynomial in 1:1x and 1:1y[ Each term of the polynomial includes one of the operators 12:1x2\ 12:"1x1 1y#\ 12:"1x 1y1# and 12:1y2[ Further\ it is found that for an operator Lij\ the terms of its polynomial are either all in 12:1x2 and 12:"1x 1y1#\ or all in 12:1y and 12:"1x1 1y#[ Therefore\ the operators Lij are divided into two groups\ one with odd power of 1:1x for all terms\ the other with odd power of 1:1y for all terms[ It is obvious that for an Lij from the _rst group\ the following relationships are valid] Lij "sin am x cos nu# −ðLij Łmn cos am x cos nu
"04a#
Lij "sin am x sin nu# −ðLij Łmn cos am x sin nu
"04b#
Lij "cos am x cos nu# ðLij Łmn sin am x cos nu
"04c#
Lij "cos am x sin nu# ðLij Łmn sin am x sin nu
"04d#
and for an Lij from the second group\ eqns "04aÐd# should be replaced by Lij "sin am x cos nu# ðLij Łmn sin am x sin nu
"05a#
Lij "sin am x sin nu# −ðLij Łmn sin am x cos nu
"05b#
Lij "cos am x cos nu# ðLij Łmn cos am x sin nu
"05c#
Lij "cos am x sin nu# −ðLij Łmn cos am x cos nu
"05d#
where the symbol ðLijŁmn is a number obtained by replacing 1:1x and 1:1y with am and bn\ respectively\ in the operator Lij[ Substituting the trigonometric series "02aÐe# into the _ve _nal equations "not presented#\ taking into account eqns "04aÐd# and "05aÐd# and executing the Galerkin procedure yield the following algebraic equations]
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"06#
us ¹ uw Þ ij ¹f ut Tij ¦f¹ uh Smn ¦f uf Þ ij ¦S" f ijmn ij Þ ij H ijmn Fmn # f ij W
"07#
vs vf Þ Smn ¦f¹ ijmn Fmn # f ijvw Wij f ijvt Tij ¦f ijvh Hij ¦S" ¹f ijmn Þ
"08#
vs vf ¹f ijvt Þ Tij ¦f¹ ijvh H Smn ¦f ijmn Fmn # ¹f ijvw W Þ ij Þ ij ¦S" f ijmn
"19#
wh ¹ ws Smn ¦f¹ wf Fmn #¦f ijww Wij ¦1SSam bn ðLŁmn him d jn W f wt Þ mn 9 ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"10#
ws ¹ ww Þ ij −1SSan bn ðLŁmn him d jn Wmn 9 ¹f wt Tij ¦f¹ ijwh H Smn ¦f wf Þ ij ¦S" f ijmn ij Þ ijmn Fmn #¦f ij W
"11#
ch ¹ cs Smn ¦f¹ wf Fmn # f ijcw Wij f ct ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"12#
cs ¹ cw Þ ij ¹f ct Tij ¦f¹ ijch H Smn ¦f cf Þ ij ¦S" f ijmn ij Þ ijmn Fmn # f ij W
"13#
fh ¹ fs Smn ¦f¹ ff Fmn # f ijfw Wij f ft ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"14#
fs ¹ fw Þ ij ¹f ft Tij ¦f¹ ijfh H Smn ¦f ff Þ ij ¦S" f ijmn ij Þ ijmn Fmn # f ij W
"15#
where i 0\ 1\ [ [ [ \ M^ j 0\ 1\ [ [ [ \ N\ and hij
1ð0−"−0# i¦j Łi 1
1
"i −j #p
\ dij
6
0 if i j 9 if i j
In the derivation of eqns "10# and "11# the assumption 1Txy
11 w 11 w ¼ 1S 1x 1y 1x 1y
has been made for linearization so that an eigenvalue problem can be obtained[ All the coe.cients in eqns "06#Ð"15# are listed in the Appendix[ Sij\ Þ Fij and Wij[ Similarly\ from From eqns "06# and "12#\ Tij and Hij can be solved in terms of Þ eqns "07# and "13#\ Þ Tij and H Þ ij[ By using these results\ Tij\ Þ ij can be solved in terms of Sij\ Fij and W Tij and H Hij\ Þ Þ ij in eqns "10#\ "11#\ "19#\ "15#\ "08# and "14# are eliminated to give ws S"C Smn ¦C Fmn #¦C ijww Wij ¦1SSam bn ðLŁmn him d jn W Þijmn Þ Þwf Þ mn 9 ijmnÞ
"16a#
ws S"C ijmn Smn ¦C wf Þifww W Þ ij −1SSam bn ðLŁmn him d jn Wmn 9 ijmn Fmn #¦C
"16b#
S
S
$ $
vs vf bijmn bijmn fs bijmn bff ijmn vs vf b¹ijmn b¹ijmn fs b¹ijmn b¹ff ijmn
Smn
b¹ijvw fw W Þ ij b¹ij
%6 7 6 7 %6 7 6 7 Fmn
Þ Smn
Fmn Þ
bijvw
bijfw
Wij
"16c#
"16d#
The expressions of the new coe.cients appearing in eqns "16aÐd# are not given here for the sake of brevity[ By introducing vectors "q# "S00 \ S10 \ [ [ [ \ SM0 \ [ [ [ \ S0N \ S1N \ [ [ [ \ SMN \ F00 \ F10 \ [ [ [ \ FM0 \ [ [ [ \ F0N \ F1N \ [ [ [ \ FMN # T
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"q¹# the same as above but with a super bar on each element[ "h# "W00 \ W10 \ [ [ [ \ WM0 \ [ [ [ \ W0N \ W1N \ [ [ [ \ WMN # T "h¹ # the same as above but with a super bar on each element[ eqns "16aÐd# can be transformed into matrix form ÞŁ"q¹#¦ðDŁ"h#¦SðE ðV ÞŁ"h¹ # 9
"17a#
ÞŁ"h¹ #−SðEŁ"h# 9 ðVŁ"q#¦ðD
"17b#
Þ Ł"h¹ # ðFŁ"q# ðC
"17c#
ðF ÞŁ"q¹# ðCŁ"h#
"17d#
Again the expressions of the coe.cient matrices in "17aÐd# are not presented[ Eliminating "q# and "q¹# in eqns "17aÐd# yields a set of equations in "h# and "h¹ #\ ðKŁ"h#¦SðE Þ Ł"h¹ # 9
"18a#
ðK Þ Ł"h¹ #−SðEŁ"h# 9
"18b#
where ÞŁ ðF ÞŁ −0 ðCŁ¦ðDŁ ðKŁ ðV Þ Ł¦ðD ÞŁ ðK Þ Ł ðVŁ ðFŁ −0 ðC Eqns "18b# and "18a# give 0 "h# ðEŁ −0 ðK Þ Ł"h¹ # S
"29#
"ðAŁ−S 1 ðIŁ#"h¹ # 9
"20#
where ðIŁ is a unit matrix and ðAŁ is de_ned by ðAŁ −ðE ÞŁ Þ Ł −0 ðKŁ ðEŁ −0 ðK Eqn "20# is the eigenvalue problem for the buckling load\ Scr\ and half of the buckling mode vector\ "h¹ cr#[ The other half\ "hcr#\ can be obtained with eqn "29#[ From eqn "20# it is found that if Scr is a buckling load\ then −Scr is a buckling load\ too[ This conclusion is in agreement with the fact that if a boundary torque causes buckling\ then the reverse of it causes buckling\ too[
3[ Numerical examples The _rst example is a laminated cylindrical shell studied by Tabiei and Simitses "0883#\ with the ply properties E00 038[508 GPa\ E11 E22 8[817 GPa\ G01 G02 3[370 GPa G12 1[440 GPa\ n01 n02 9[17\ n12 9[34
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and the stacking sequence "9>:89>:9># s[ It is obvious that the more terms are used in the truncated series "02aÐe#\ the better results can be obtained and larger amount of computation is needed[ The study of convergence indicates that M 5 and N 09 can guarantee that the buckling load to be obtained has more than four signi_cant digits for all examples studied in this Section[ Let S cr and Scr denote\ respectively\ the buckling loads based on the _rst!order shear deformation theory with and without a shear correction factor of 4:5[ For R:h 099\ L:R 0 and h 9[990894 m\ the calculation gives the buckling loads "in 095 N:m# S cr 9[0280"7# and Scr 9[0282"7#\ where the number in parentheses is the circumferential wave number[ By comparison with Scr 9[0457"8# given by Tabiei and Simitses "0883# based on the completely clamped boundary conditions\ it can be seen that the mixed boundary conditions yield appreciably lower buckling load and less circumferential wave number than the completely clamped boundary conditions[ However\ for long shells\ the di}erence is negligible[ For instance\ for R:h 099 and L:R 4\ the present theory gives buckling loads S cr Scr 9[9640"4# which is almost equal to Scr 9[9646"5# given by Tabiei and Simitses "0883#[ The full pro_le of Scr vs L:R is shown in Fig[ 0[ It may be interesting to take a look at the buckling mode[ First it is found that the buckling mode actually has a single wavelength in the circumferential direction while it is a combination of many waves of di}erent wavelengths in the axial direction[ For instance\ for R:h 099 and L:R 0\ the buckling mode is wcr "−41[82 sin a0 x¦62[00 sin a1 x¦4[915 sin a2 x¦= = =# cos 4u ¦"04[60 sin a0 x¦135[3 sin a1 x−0[381 sin a2 x¦= = =# sin 4u in which all terms with the circumferential wave number n 4 disappear because they are exactly zero[ This _nding suggests an alternative way to calculate the buckling load[ By _xing the number n in series "02aÐe#^ instead of taking summation for it\ the procedure developed in the preceding sections becomes simpler[ For each speci_ed n\ the simpli_ed procedure gives a buckling load Sn[
Fig[ 0[ The curve of Scr vs L:R at R:h 099 for Example 0 is based on mixed boundary conditions[ The stars are results from Tabiei and Simitses "0883# based on completely clamped boundary conditions[
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As usual\ the lowest Sn for all n is the actual buckling load "critical load# Scr\ and the number of n which yields Scr is the actual circumferential wave number of the buckling mode[ But here in this paper the authors prefer presenting the theory with double trigonometric series\ instead of single trigonometric series obtained by _xing n\ because in the form of double trigonometric series the theory can be extended to cover more general cases such as postbuckling[ The second detail about the buckling mode worth mentioning is that for each buckling load\ the computer gives two di}erent buckling modes[ This means that the corresponding eigenvalue is a twofold repeated root[ As a matter of fact\ the two buckling modes can be obtained from each other by suitable shift of the origin of the coordinate u\ which does not a}ect the buckling load[ To show this\ let wc0 "a0 sin a0 x¦a1 sin a1 x¦= = =# cos pu¦"a¹0 sin a0 x¦a¹1 sin a1 x¦= = =# sin pu be a buckling mode[ The shift of the origin of u by j:p makes cos pu and sin pu be replaced by cos"pu−j# and sin"pu−j#\ respectively\ and wc0 be transformed to another mode wc1 ð"a0 cos j−a¹0 sin j# sin a0 x¦"a1 cos j−a¹1 sin j# sin a1 x¦= = =Ł cos pu ¦ð"a0 sin j¦a¹0 cos j# sin a0 x¦"a1 sin j¦a¹1 cos j# sin a1 x¦= = =Ł sin pu Thus\ in_nitely many such buckling modes can be obtained by assigning di}erent values to j[ But the number of linearly independent modes is at most two[ Therefore\ for each buckling load the computer gives only two di}erent modes\ not more\ which represent the same actual physical mode[ Another example is a clamped unsymmetrically laminated shell cited from Hui and Du "0876#[ The stacking sequence is "89> in:9> out#\ with the ply properties E00 :E11 09[9\ G01 :E11 9[4\ n01 9[14\ n12 9[3\ G12 :E11 9[16 For R:h 099 and L:R 0 the present theory gives the dimensionless critical shear stress tcr
Scr R E11 T 1
9[3885"8#
while Hui and Du "0876# gives tcr 9[4916"8[3#[ The two results are quite close in both buckling load and wave number[ But for a shorter shell\ say L:R 9[4\ the two results tcr 9[7906"09#
from the present theory
tcr 9[8602"09#
from Hui and Du "0876#
are appreciably di}erent[ Comparison for various R:h is shown in Fig[ 1[
4[ Conclusions A new e.cient method is developed in this paper to deal with mixed boundary conditions for transversely shear deformable cylindrical shells[ The main idea is to take the displacements and forces related to the boundary conditions as basic unknowns and to express those not related to the boundary conditions in terms of the basic unknowns by taking inverse of a matrix composed
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Fig[ 1[ The relationship between Scr and R:h at L:R 0 for Example 1[ The thick curve is based on mixed boundary conditions[ The thin curve is based on completely clamped boundary conditions from Hui and Du "0876#[
of operators[ The numerical results of two torsional buckling problems of such shells show that for short shells the present method makes improvement in calculation of buckling loads and wave numbers compared with the methods in the literature which are based on fully clamped boundaries[
Appendix The coe.cients in eqns "06#Ð"15# are listed below] us ¹f ijmn f uh bn "di\m−0 −di\m¦0 #d jn \ ¹f uf f ijuw 9 f ut ij ai \ ij 9\ ijmn 9\ ut uh us us ¹f uh ¹ uf ¹f ijuw f ijuw ¹f ut f ijmn −f¹ ijmn \ f uf ij f ij \ ij f ij \ ijmn −f ijmn \
¹f ijvt A01 ai b j ðL00 Łij ¦"A11 b1j ¦A33 :R 1 #ðL10 Łij ¦B01 ai b j ðL20 Łij ¦"B11 b1j −A33 :R#ðL30 Łij ¹f ijvh A01 ai b j ðL01 Łij ¦"A11 b1j ¦A33 :R 1 #ðL11 Łij ¦B01 ai b j ðL21 Łij ¦"B11 b1j −A33 :R#ðL31 Łij vs f ijmn −"am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn −A01 bn "am−0 ðL02 Łm−0\n di\m−0 −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn −"A11 bn1 ¦A33 :R 1 #"ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn −B01 bn "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn −"B11 bn1 −A33 :R#"ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724 vf f ijmn −A01 bn "am−0 ðL03 Łm−0\n di\m−0 −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn
−"A11 bn1 ¦A33 :R 1 #"ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn −B01 bn "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn −"B11 bn1 −A33 :R#"ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\n di\m¦0 #d jn ¹f ijvw −
A11 ¦A33 A01 vt B01 vh ¹f ¦ ¹f b j ðLŁij ¦ R R ij R ij
vs vs vf vf f ijvt −f¹ ijvt \ f ijvh −f¹ ijvh \ ¹f ijmn f ijmn \ ¹f ijmn f ijmn \ f ijvw −f¹ ijvw
f wt ij − − f ijwh − − ws ¹f ijmn −
− −
−
¹f wf ijmn − − −
−
A01 A11 ¦A33 a ðL Ł − b j ðL10 Łij R i 00 ij R
0
1
0
1
B01 B11 −A44 ai ðL20 Łij − −A33 b j ðL30 Łij R R
A01 A11 ¦A33 a ðL Ł − b j ðL11 Łij R i 01 ij R
0
1
0
1
B01 B11 −A44 ai ðL21 Łij − −A33 b j ðL31 Łij R R
A01 ðL Ł d −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn "a R m−0 02 m−0\n i\m−0 A11 ¦A33 bn "ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn R
0 0
1 1
B01 −A44 "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn R B11 −A33 bn "ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn R
A01 ðL Ł d −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn "a R m−0 03 m−0\n i\m−0 A11 ¦A33 bn "ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn R
0 0
1 1
B01 −A44 "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn R B11 −A33 bn "ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\n di\m¦0 #d jn R
2722
2723
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0
f ijww − A44 ai1 ¦A33 b1j ¦
A11 R
1
1
ðLŁij −
A01 wt B01 wh f − f R ij R ij
wt ws ws ¹f wt ¹f ijwh f ijwh \ f ijmn ¹ wf ¹f ijww f ijww −f¹ ijmn \ f wf ij f ij \ ijmn −f ijmn \
f ct ij A44 ðL20 Łij f ijch ai ðLŁij ¦A44 ðL21 Łij cs ¹f ijmn A44 "ðL22 Łm−0\n di\m−0 −ðL22 Łm¦0\n di\m¦0 #d jn
¹f cf ijmn ðbn "ðLŁm−0\n di\m−0 −ðLŁm¦0\n di\m¦0 # ¦A44 "ðL23 Łm−0\n di\m−0 −ðL23 Łm¦0\n di\m¦0 #Łd jn f ijcw A44 ai ðLŁij ¦
A01 ct B01 ch f ¦ " f −ai ðLŁij # R ij R ij
ct cs cs ¹f ct ¹f ijch f ijch \ f ijmn ¹ cf ¹f ijcw f ijcw −f¹ ijmn \ f cf ij f ij \ ijmn −f ijmn \
0
¹f ft ij A01 ¦
1
0
¦ B01 ¦
0
¹f ijfh A01 ¦
0
1
B01 B11 1 ai b j ðL00 Łij ¦ A11 ¦ b j ðL10 Łij R R
1
0
1
D01 D11 1 ai b j ðL20 Łij ¦ B11 ¦ b j ðL30 Łij R R
1
0
1
B01 B11 1 ai b j ðL01 Łij ¦ A11 ¦ b j ðL11 Łij R R
0
¦ B01 ¦
1
0
1
D01 D11 1 ai b j ðL21 Łij ¦ B11 ¦ b j ðL31 Łij R R
fs f ijmn −"am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn
0 0 0 0
1 1 1 1
− A01 ¦
B01 bn "am−0 ðL02 Łm−0\n di\m−0 −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn R
− A11 ¦
B11 1 bn "ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn R
− B01 ¦
D01 bn "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn R
− B11 ¦
D11 1 bn "ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn R
0 fs f ijmn − "am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn R
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0 0 0 0
1 1 1 1
− A01 ¦
B01 bn "am−0 ðL03 Łm−0\n di\m−0 −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn R
− A11 ¦
B11 1 bn "ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn R
− B01 ¦
D01 bn "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn R
− B11 ¦
D11 1 bn "ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\m di\m¦0 #d jn R
¹f ijfw −
0
2724
1
0 B11 A01 ft B01 fh ¹f ¦ ¹f A11 ¦ b j ðLŁij ¦ R R R ij R ij
fs fs ff ¹ ft f ijfh −f¹ ijfh \ ¹f ijmn f ft f ijmn \ ¹f ff f ijfw −f¹ ijfw ij −f ij \ ijmn f ijmn \
References Flugge\ W[\ 0859[ Stresses in Shells[ Springer!Verlag\ Berlin\ Gottingen\ Heidelberg[ Hui\ D[\ Du\ I[H[Y[\ 0876[ Initial postbuckling behavior of imperfect antisymmetric cross!ply cylindrical shells under torsion[ ASME J[ Appl[ Mech[ 43\ 063Ð079[ Simitses\ G[J[\ Shaw\ D[\ 0874[ Imperfection sensitivity of laminated cylindrical shells in torsion and axial compression[ Composite Structures 3\ 224Ð259[ Stein\ M[\ 0875[ Nonlinear theory for plates and shells including the e}ects of transverse shearing[ AIAA J[ 13\ 0426Ð 0433[ Tabiei\ A[\ Simitses\ G[J[\ 0883[ Buckling of moderately thick\ laminated cylindrical shells under torsion[ AIAA J[ 21\ 528Ð536[
International Journal of Solids and Structures 25 "0888# 2710Ð2724
Buckling analysis of a laminated cylindrical shell under torsion subjected to mixed boundary conditions Renjie Maoa\ \ C[!H[ Lub a
b
Department of En`ineerin` Mechanics\ Shan`hai Jiao Ton` University\ Shan`hai 199929\ P[R[ China R+D Senior En`ineer\ Mechanical Systems\ Dover Elevator Systems\ Inc[\ Horn Lake\ MS 27526\ U[S[A[ Received 17 October 0886^ in revised form 11 May 0887
Abstract A new e.cient method is developed in this paper for buckling analysis of a cross!ply laminated cylindrical shell under torsion subjected to mixed boundary conditions[ The transverse shear is taken into account by a _rst!order theory with a shear correction factor of 4:5[ The mixed boundary conditions include conditions in forces as well as conditions in displacements\ and these forces and displacements are selected as basic unknowns[ The other displacements and forces are expressed in terms of the basic unknowns by taking inverse of a matrix composed of operators[ The equations of buckled equilibrium in terms of the basic unknowns are solved with double trigonometric series which satisfy the mixed boundary conditions[ Com! parison of the obtained numerical results with those given in the literature based on completely clamped boundary conditions checks with the fact that the mixed boundary conditions yield appreciably lower buckling load and less circumferential wave number than the completely clamped boundary conditions[ The curves in the _gures show how the di}erence in buckling loads between the two kinds of boundary conditions varies when the length and thickness of the shell vary[ Þ 0888 Elsevier Science Ltd[ All rights reserved[
0[ Introduction The buckling of cylindrical shells under torsion "torsional buckling# has been much less studied in contrast to the buckling under axial compression "compressive buckling#\ especially for anisotropic shells[ The main di.culty involved in the torsional buckling is in that _rstly the nonlinear terms in the equations of buckled equilibrium include partial derivatives both in axial direction and in circumferential direction\ so that the single!wave buckling mode\ which is useful in the analysis of compressive buckling\ does not work\ and secondly the boundary conditions in torsional force Txy and in torsional moment Mxy can hardly be satis_ed beforehand by the assumed buckling mode[ Simitses and Shaw "0874# and Hui and Du "0876# solve the di.culty for Donnell!type shallow
Corresponding author[ 9919Ð6572:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII ] S 9 9 1 9 Ð 6 5 7 2 " 8 7 # 9 9 0 6 7 Ð 3
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R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
shells without transverse shear by introducing a stress function and assuming a multi!wave buckling mode such as truncated trigonometric series in u with the coe.cients being functions of x\ where x is the axial coordinate in length and u is the circumferential coordinate in angle[ Thus the equations of equilibrium and compatibility reduce to ordinary di}erential equations for the coe.cients of the series\ and the original boundary conditions yield boundary conditions for the solution of these ordinary di}erential equations[ In principle\ the above solution procedure may apply to non!shallow shells with transverse shear[ But in this case the number of the unknowns will be _ve and\ due to the use of non!shallow shell theory\ no simple stress function can be introduced to reduce that number[ The above solution procedure will then be much more involved[ To solve the problem\ Tabiei and Simitses "0883# express the equations of equilibrium in terms of the three displacements and two rotations and solve them with double trigonometric series which vanish at the boundaries[ Tabiei and Simitses| method is useful when all boundary conditions are in displacements and rotations\ i[e[\ the shell is completely clamped at its ends\ or when the shell is long enough so that the boundary conditions play little role[ However\ in the case of mixed boundary conditions\ some of the conditions are given as boundary forces[ They can hardly be satis_ed beforehand by the double trigonometric series for displacements and rotations assumed by Tabiei and Simitses "0883#[ That is why the torsional buckling of transversely shear deformable shells under mixed boundary conditions has not yet been studied much[ In many practical cases\ however\ the real physical boundaries may be described better by mixed boundary conditions than by completely clamped boundary conditions[ Therefore\ solution methods are needed to deal with mixed boundary conditions[ The present study undertakes this task to develop such a method[ In the present study\ the displacements "including rotations# and forces "including moments# appearing in the mixed boundary conditions are selected as basic unknowns and are expressed in terms of di}erent kinds of double trigonometric series which satisfy the mixed boundary conditions[ Then\ by taking inverse of a matrix composed of operators\ all other displacements and forces\ and hence the _ve equations of buckled equilibrium\ can be expressed in terms of the _ve basic unknowns[ Finally\ the application of the Galerkin procedure to the equations of buckled equilibrium leads to an eigenvalue problem[ The numerical results of examples show that the mixed boundary conditions lead to lower buckling load and less cir! cumferential wave number of the buckling mode than the completely clamped boundary conditions[
1[ Basic equations A thin cross!ply cylindrical shell subjected to torsion is under consideration[ In the _rst!order shear!deformation theory the displacement _eld of a cylindrical shell is ux u¦zc\ uy v¦z8\ uz w
"0#
where "x\ y\ z# are\ respectively\ the axial\ circumferential and normal coordinates in length\ "ux\ uy\ uz# are their corresponding displacement components at an arbitrary point of the shell\ and "u\ v\ w# are those components at the middle surface[ The linear strains at an arbitrary point of the shell can be derived from the displacement _eld\
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0
1u 1v w 1u 1v 1c 18 1c 18 ¦z \ o¹y ¦ ¦z \ g¹ xy ¦ ¦z ¦ 1x 1x 1y R 1y 1y 1x 1y 1x
o¹x
1
1w 1w y ¦c\ g¹ yz − ¦8 1x 1y R
g¹ xz
2712
"1#
These strains coincide with the general _rst!order shear!deformation kinematic relations of Tabiei and Simitses "0883# for linear case\ if the thin shell assumption z:R ð 0 is taken into account[ The de_nitions of the stress resultants are as usual
g
g
g g
g
Tx sx dz\ Ty sy dz\ Mx sx z dz\ My sy z dz
g
Txy Tyx txy dz\ Mxy Myx txy z dz
g
g
Qx txz dz\ Qy tyz dz
"2#
where
g g
h:1
\ h is the thickness of the shell
−h:1
By introducing vectors "s# "Tx \ Ty \ Txy \ Mx \ My \ Mxy \ Qy \ Qx # T "o# "ox \ oy \ gxy \ kx \ ky \ kxy \ gyz \ gxz # T where ox
1u 1v w 1u 1v \ oy ¦ \ gxy ¦ 1x 1y R 1y 1x
kx
1c 18 1c 18 \ ky \ kxy ¦ \ gyz g¹ yz \ gxz g¹ xz 1x 1y 1y 1x
are the strains and curvature changes of the middle surface\ the constitutive equations for cross! ply laminates can be written in matrix form "s# ðCŁ"o#
"3#
where
&
ðAŁ
ðBŁ
ðCŁ ðBŁ
ðDŁ ðA Ł
'
&
A00 A01
\
' &
B00 B01
ðAŁ A01 A11
\
A55
ðBŁ B01 B11 B55
'
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R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
&
D00 D01
ðDŁ D01 D11 D55
'
\
ðA Ł
$
A33 A44
%
The equations of equilibrium developed by Stein "0875# are used in the present study[ For a thin shell\ after imposing the following three assumptions about the nonlinear terms] "a# The only load is the boundary torques\ so that all nonlinear terms not related to the torsional force Txy are neglected^ "b# All nonlinear terms not related to w or its derivatives are neglected^ "c# All nonlinear terms not in the equation of normal equilibrium are neglected^ these equations reduce to 1Tx 1Txy ¦ 9 1x 1y
"4a#
1Txy 1Ty Qy ¦ ¦ 9 1x 1y R
"4b#
11 w 1Qx 1Qy Ty ¦ − ¦1Txy 9 1x 1y R 1x 1y
"4c#
1Mx 1Mxy ¦ −Qx 9 1x 1y
"4d#
1Mxy 1My ¦ −Qy 9 1x 1y
"4e#
and the variationally consistent boundary conditions can be written in the form "Tx du#= 9L 9
"5a#
ð"Txy −S#dvŁ= 9L 9
"5b#
$0
Txy
L
1 %b
1w ¦Qx dw 1y
9
"5c#
9
"Mx dc#= 9L 9
"5d#
"Mxy d8#= 9L 9
"5e#
"Txy du#= 91p 9
"6a#
"Ty dv#= 91p 9
"6b#
$0
Txy
1p
1 %b
1w ¦Qy dw 1x
9
9
"6c#
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
2714
"Mxy dc#= 91p 9
"6d#
"My d8#= 91p 9
"6e#
The symbols = 9L and = 91p are de_ned as "Tx du#= 9L "Tx du# x L −"Tx du# x 9 \ and so on^ "Txy du#= 91p "Txy du# u 1p −"Txy du# u 9 \ and so on where L and R are\ respectively\ the length and radius of the shell\ u y:R and S is the torsional force per unit length of y produced by the boundary torques[ Conditions "6aÐe# can be satis_ed by the requirement that all displacements and forces appearing in these conditions be periodic in u with a period equal to 1p[ For conditions "5aÐe#\ the following mixed boundary conditions are assumed Tx 9\ Txy S\ w 9\ Mx Mxy 9 at x 9\ L
"7#
It is obvious that conditions "5aÐe# will be satis_ed if conditions "7# are satis_ed[ Equilibrium equations "4aÐe# and boundary conditions "5a#Ð"6e# coincide with those of Tabiei and Simitses "0883# for thin shells using the _rst!order shear!deformation kinematic relations[ The equations of Tabiei and Simitses "0883# is based on Donnell!type theory[ This suggests that the error caused by the preceding assumptions "a#Ð"c# can be expected not to be larger than that of Donnell!type theory[ More complete equations of equilibrium for a thin cylindrical shell are given by Flugge "0859#[ These equations correspond to a system of kinematic relations in which all nonlinear terms of u\ v and w are retained[
2[ Solution The _ve variables Tx\ Txy\ w\ Mx and Mxy appearing in the mixed boundary conditions "7# are selected as basic unknowns[ The equations of buckled equilibrium "4aÐe# can be transformed into a system of _ve equations in the basic unknowns by eliminating all other displacements and forces[ To do this\ the displacements and rotations u\ v\ c and 8 are expressed in terms of the basic unknowns by using eqn "3# to obtain F u J F Tx −A01 w:R J G G G G v f jMx −B01 w:Rf j ðGŁ J F J F Txy c G G G G Mxy j f 8j f where
"8#
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R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
K 1 1 1 1L HA00 A01 B00 B01 H 1x 1y 1x 1y H H H 1 1 d 1H HB00 B01 D00 D01 H 1x 1y dx 1yH ðGŁ H H 1 1 1 1H A55 B55 B55 H HA55 1y 1x 1y 1x H H H 1 1 1 1H B55 D55 D55 H H B55 1y 1x 1y 1xl k
Let Gij represent the elements of ðGŁ\ Lij be the cofactor of the element Gji\ and L be the determinant of ðGŁ[ They are all operators[ The inverse of ðGŁ can be written as ðGŁ −0 L−0 ðLij Ł
"09#
Substitution of eqn "09# into eqn "8# gives F Tx −A01 w:R J FuJ G G G G Mx −B01 w:R f vf j j L J F ðLij Ł J F Txy c G G G G Mxy f j f8j
"00#
The stress resultants Ty\ My\ Qx and Qy can then be expressed in terms of the basic unknowns through the constitutive equations] Ty A01
Qy A33
1 1
"01a#
1u 1v w 1c 18 ¦B11 ¦ ¦D01 ¦D11 1x 1y R 1x 1y
"01b#
My B01 Qx A44
0 0 1
1u 1v w 1c 18 ¦A11 ¦ ¦B01 ¦B11 1x 1y R 1x 1y
0 0
1w ¦c 1x
1w v − ¦8 1y R
"01c#
1
"01d#
By application of the operator L to each side of eqns "4bÐe# and using eqns "01aÐd# and eqn "00# to eliminate Ty\ My\ Qx and Qy\ eqns "4aÐe# can be transformed into a system of _ve _nal equations in the _ve basic unknowns[ These _nal equations are too lengthy to be presented here[ The _ve _nal equations are solved with truncated double trigonometric series Tx SSF1sin am x"Tmn cos nu¦T Þmn sin nu#
"02a#
Txy S¦S"cos am−0 x−cos am¦0 x#"Smn cos nu¦S Þmn sin nu#
"02b#
w S sin am x"Wmn cos nu¦W Þ mn sin nu#
"02c#
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
2716
Mx S sin am x"Hmn cos nu¦H Þ mn sin nu#
"02d#
Mxy S"cos am−0 x−cos am¦0 x#"Fmn cos nu¦F Þmn sin nu#
"02e#
where\ and in the following\ the symbols M
M
S s s \ am m 0 n 0
mp L
Tmn\ [ [ [ \ Þ Fmn are to be determined[ are used for brevity in writing[ The constant coe.cients Tmn\ Þ Trigonometric series "02# satisfy the mixed boundary conditions "7#[ Before using the Galerkin procedure\ a closer inspection of the operators in eqns "00# is useful[ The operator L is a fourth!order homogeneous polynomial in 1:1x and 1:1y[ Each term of the polynomial includes one of the operators 13:1x3\ 13:"1x1 1y1# and 13:1y3[ Therefore\ when the operator L is applied to a product of trigonometric functions such as sin amx cos nu\ a very simple result can be obtained L"sin am x cos nu# ðLŁmn sin am x cos nu
"03#
where the symbol ðLŁmn is a number denoting the determinant of a matrix obtained by replacing 1:1x and 1:1y with am and bn\ respectively\ in the operator L\ where bn n:R[ Each of the operators Lij "i\ j 0\ 1\ 2\ 3# is a third!order polynomial in 1:1x and 1:1y[ Each term of the polynomial includes one of the operators 12:1x2\ 12:"1x1 1y#\ 12:"1x 1y1# and 12:1y2[ Further\ it is found that for an operator Lij\ the terms of its polynomial are either all in 12:1x2 and 12:"1x 1y1#\ or all in 12:1y and 12:"1x1 1y#[ Therefore\ the operators Lij are divided into two groups\ one with odd power of 1:1x for all terms\ the other with odd power of 1:1y for all terms[ It is obvious that for an Lij from the _rst group\ the following relationships are valid] Lij "sin am x cos nu# −ðLij Łmn cos am x cos nu
"04a#
Lij "sin am x sin nu# −ðLij Łmn cos am x sin nu
"04b#
Lij "cos am x cos nu# ðLij Łmn sin am x cos nu
"04c#
Lij "cos am x sin nu# ðLij Łmn sin am x sin nu
"04d#
and for an Lij from the second group\ eqns "04aÐd# should be replaced by Lij "sin am x cos nu# ðLij Łmn sin am x sin nu
"05a#
Lij "sin am x sin nu# −ðLij Łmn sin am x cos nu
"05b#
Lij "cos am x cos nu# ðLij Łmn cos am x sin nu
"05c#
Lij "cos am x sin nu# −ðLij Łmn cos am x cos nu
"05d#
where the symbol ðLijŁmn is a number obtained by replacing 1:1x and 1:1y with am and bn\ respectively\ in the operator Lij[ Substituting the trigonometric series "02aÐe# into the _ve _nal equations "not presented#\ taking into account eqns "04aÐd# and "05aÐd# and executing the Galerkin procedure yield the following algebraic equations]
2717
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724 uh ¹ us Smn ¦f¹ uf f ut Fmn # f ijuw Wij ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"06#
us ¹ uw Þ ij ¹f ut Tij ¦f¹ uh Smn ¦f uf Þ ij ¦S" f ijmn ij Þ ij H ijmn Fmn # f ij W
"07#
vs vf Þ Smn ¦f¹ ijmn Fmn # f ijvw Wij f ijvt Tij ¦f ijvh Hij ¦S" ¹f ijmn Þ
"08#
vs vf ¹f ijvt Þ Tij ¦f¹ ijvh H Smn ¦f ijmn Fmn # ¹f ijvw W Þ ij Þ ij ¦S" f ijmn
"19#
wh ¹ ws Smn ¦f¹ wf Fmn #¦f ijww Wij ¦1SSam bn ðLŁmn him d jn W f wt Þ mn 9 ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"10#
ws ¹ ww Þ ij −1SSan bn ðLŁmn him d jn Wmn 9 ¹f wt Tij ¦f¹ ijwh H Smn ¦f wf Þ ij ¦S" f ijmn ij Þ ijmn Fmn #¦f ij W
"11#
ch ¹ cs Smn ¦f¹ wf Fmn # f ijcw Wij f ct ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"12#
cs ¹ cw Þ ij ¹f ct Tij ¦f¹ ijch H Smn ¦f cf Þ ij ¦S" f ijmn ij Þ ijmn Fmn # f ij W
"13#
fh ¹ fs Smn ¦f¹ ff Fmn # f ijfw Wij f ft ij Tij ¦f ij Hij ¦S" f ijmnÞ ijmnÞ
"14#
fs ¹ fw Þ ij ¹f ft Tij ¦f¹ ijfh H Smn ¦f ff Þ ij ¦S" f ijmn ij Þ ijmn Fmn # f ij W
"15#
where i 0\ 1\ [ [ [ \ M^ j 0\ 1\ [ [ [ \ N\ and hij
1ð0−"−0# i¦j Łi 1
1
"i −j #p
\ dij
6
0 if i j 9 if i j
In the derivation of eqns "10# and "11# the assumption 1Txy
11 w 11 w ¼ 1S 1x 1y 1x 1y
has been made for linearization so that an eigenvalue problem can be obtained[ All the coe.cients in eqns "06#Ð"15# are listed in the Appendix[ Sij\ Þ Fij and Wij[ Similarly\ from From eqns "06# and "12#\ Tij and Hij can be solved in terms of Þ eqns "07# and "13#\ Þ Tij and H Þ ij[ By using these results\ Tij\ Þ ij can be solved in terms of Sij\ Fij and W Tij and H Hij\ Þ Þ ij in eqns "10#\ "11#\ "19#\ "15#\ "08# and "14# are eliminated to give ws S"C Smn ¦C Fmn #¦C ijww Wij ¦1SSam bn ðLŁmn him d jn W Þijmn Þ Þwf Þ mn 9 ijmnÞ
"16a#
ws S"C ijmn Smn ¦C wf Þifww W Þ ij −1SSam bn ðLŁmn him d jn Wmn 9 ijmn Fmn #¦C
"16b#
S
S
$ $
vs vf bijmn bijmn fs bijmn bff ijmn vs vf b¹ijmn b¹ijmn fs b¹ijmn b¹ff ijmn
Smn
b¹ijvw fw W Þ ij b¹ij
%6 7 6 7 %6 7 6 7 Fmn
Þ Smn
Fmn Þ
bijvw
bijfw
Wij
"16c#
"16d#
The expressions of the new coe.cients appearing in eqns "16aÐd# are not given here for the sake of brevity[ By introducing vectors "q# "S00 \ S10 \ [ [ [ \ SM0 \ [ [ [ \ S0N \ S1N \ [ [ [ \ SMN \ F00 \ F10 \ [ [ [ \ FM0 \ [ [ [ \ F0N \ F1N \ [ [ [ \ FMN # T
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
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"q¹# the same as above but with a super bar on each element[ "h# "W00 \ W10 \ [ [ [ \ WM0 \ [ [ [ \ W0N \ W1N \ [ [ [ \ WMN # T "h¹ # the same as above but with a super bar on each element[ eqns "16aÐd# can be transformed into matrix form ÞŁ"q¹#¦ðDŁ"h#¦SðE ðV ÞŁ"h¹ # 9
"17a#
ÞŁ"h¹ #−SðEŁ"h# 9 ðVŁ"q#¦ðD
"17b#
Þ Ł"h¹ # ðFŁ"q# ðC
"17c#
ðF ÞŁ"q¹# ðCŁ"h#
"17d#
Again the expressions of the coe.cient matrices in "17aÐd# are not presented[ Eliminating "q# and "q¹# in eqns "17aÐd# yields a set of equations in "h# and "h¹ #\ ðKŁ"h#¦SðE Þ Ł"h¹ # 9
"18a#
ðK Þ Ł"h¹ #−SðEŁ"h# 9
"18b#
where ÞŁ ðF ÞŁ −0 ðCŁ¦ðDŁ ðKŁ ðV Þ Ł¦ðD ÞŁ ðK Þ Ł ðVŁ ðFŁ −0 ðC Eqns "18b# and "18a# give 0 "h# ðEŁ −0 ðK Þ Ł"h¹ # S
"29#
"ðAŁ−S 1 ðIŁ#"h¹ # 9
"20#
where ðIŁ is a unit matrix and ðAŁ is de_ned by ðAŁ −ðE ÞŁ Þ Ł −0 ðKŁ ðEŁ −0 ðK Eqn "20# is the eigenvalue problem for the buckling load\ Scr\ and half of the buckling mode vector\ "h¹ cr#[ The other half\ "hcr#\ can be obtained with eqn "29#[ From eqn "20# it is found that if Scr is a buckling load\ then −Scr is a buckling load\ too[ This conclusion is in agreement with the fact that if a boundary torque causes buckling\ then the reverse of it causes buckling\ too[
3[ Numerical examples The _rst example is a laminated cylindrical shell studied by Tabiei and Simitses "0883#\ with the ply properties E00 038[508 GPa\ E11 E22 8[817 GPa\ G01 G02 3[370 GPa G12 1[440 GPa\ n01 n02 9[17\ n12 9[34
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and the stacking sequence "9>:89>:9># s[ It is obvious that the more terms are used in the truncated series "02aÐe#\ the better results can be obtained and larger amount of computation is needed[ The study of convergence indicates that M 5 and N 09 can guarantee that the buckling load to be obtained has more than four signi_cant digits for all examples studied in this Section[ Let S cr and Scr denote\ respectively\ the buckling loads based on the _rst!order shear deformation theory with and without a shear correction factor of 4:5[ For R:h 099\ L:R 0 and h 9[990894 m\ the calculation gives the buckling loads "in 095 N:m# S cr 9[0280"7# and Scr 9[0282"7#\ where the number in parentheses is the circumferential wave number[ By comparison with Scr 9[0457"8# given by Tabiei and Simitses "0883# based on the completely clamped boundary conditions\ it can be seen that the mixed boundary conditions yield appreciably lower buckling load and less circumferential wave number than the completely clamped boundary conditions[ However\ for long shells\ the di}erence is negligible[ For instance\ for R:h 099 and L:R 4\ the present theory gives buckling loads S cr Scr 9[9640"4# which is almost equal to Scr 9[9646"5# given by Tabiei and Simitses "0883#[ The full pro_le of Scr vs L:R is shown in Fig[ 0[ It may be interesting to take a look at the buckling mode[ First it is found that the buckling mode actually has a single wavelength in the circumferential direction while it is a combination of many waves of di}erent wavelengths in the axial direction[ For instance\ for R:h 099 and L:R 0\ the buckling mode is wcr "−41[82 sin a0 x¦62[00 sin a1 x¦4[915 sin a2 x¦= = =# cos 4u ¦"04[60 sin a0 x¦135[3 sin a1 x−0[381 sin a2 x¦= = =# sin 4u in which all terms with the circumferential wave number n 4 disappear because they are exactly zero[ This _nding suggests an alternative way to calculate the buckling load[ By _xing the number n in series "02aÐe#^ instead of taking summation for it\ the procedure developed in the preceding sections becomes simpler[ For each speci_ed n\ the simpli_ed procedure gives a buckling load Sn[
Fig[ 0[ The curve of Scr vs L:R at R:h 099 for Example 0 is based on mixed boundary conditions[ The stars are results from Tabiei and Simitses "0883# based on completely clamped boundary conditions[
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
2720
As usual\ the lowest Sn for all n is the actual buckling load "critical load# Scr\ and the number of n which yields Scr is the actual circumferential wave number of the buckling mode[ But here in this paper the authors prefer presenting the theory with double trigonometric series\ instead of single trigonometric series obtained by _xing n\ because in the form of double trigonometric series the theory can be extended to cover more general cases such as postbuckling[ The second detail about the buckling mode worth mentioning is that for each buckling load\ the computer gives two di}erent buckling modes[ This means that the corresponding eigenvalue is a twofold repeated root[ As a matter of fact\ the two buckling modes can be obtained from each other by suitable shift of the origin of the coordinate u\ which does not a}ect the buckling load[ To show this\ let wc0 "a0 sin a0 x¦a1 sin a1 x¦= = =# cos pu¦"a¹0 sin a0 x¦a¹1 sin a1 x¦= = =# sin pu be a buckling mode[ The shift of the origin of u by j:p makes cos pu and sin pu be replaced by cos"pu−j# and sin"pu−j#\ respectively\ and wc0 be transformed to another mode wc1 ð"a0 cos j−a¹0 sin j# sin a0 x¦"a1 cos j−a¹1 sin j# sin a1 x¦= = =Ł cos pu ¦ð"a0 sin j¦a¹0 cos j# sin a0 x¦"a1 sin j¦a¹1 cos j# sin a1 x¦= = =Ł sin pu Thus\ in_nitely many such buckling modes can be obtained by assigning di}erent values to j[ But the number of linearly independent modes is at most two[ Therefore\ for each buckling load the computer gives only two di}erent modes\ not more\ which represent the same actual physical mode[ Another example is a clamped unsymmetrically laminated shell cited from Hui and Du "0876#[ The stacking sequence is "89> in:9> out#\ with the ply properties E00 :E11 09[9\ G01 :E11 9[4\ n01 9[14\ n12 9[3\ G12 :E11 9[16 For R:h 099 and L:R 0 the present theory gives the dimensionless critical shear stress tcr
Scr R E11 T 1
9[3885"8#
while Hui and Du "0876# gives tcr 9[4916"8[3#[ The two results are quite close in both buckling load and wave number[ But for a shorter shell\ say L:R 9[4\ the two results tcr 9[7906"09#
from the present theory
tcr 9[8602"09#
from Hui and Du "0876#
are appreciably di}erent[ Comparison for various R:h is shown in Fig[ 1[
4[ Conclusions A new e.cient method is developed in this paper to deal with mixed boundary conditions for transversely shear deformable cylindrical shells[ The main idea is to take the displacements and forces related to the boundary conditions as basic unknowns and to express those not related to the boundary conditions in terms of the basic unknowns by taking inverse of a matrix composed
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Fig[ 1[ The relationship between Scr and R:h at L:R 0 for Example 1[ The thick curve is based on mixed boundary conditions[ The thin curve is based on completely clamped boundary conditions from Hui and Du "0876#[
of operators[ The numerical results of two torsional buckling problems of such shells show that for short shells the present method makes improvement in calculation of buckling loads and wave numbers compared with the methods in the literature which are based on fully clamped boundaries[
Appendix The coe.cients in eqns "06#Ð"15# are listed below] us ¹f ijmn f uh bn "di\m−0 −di\m¦0 #d jn \ ¹f uf f ijuw 9 f ut ij ai \ ij 9\ ijmn 9\ ut uh us us ¹f uh ¹ uf ¹f ijuw f ijuw ¹f ut f ijmn −f¹ ijmn \ f uf ij f ij \ ij f ij \ ijmn −f ijmn \
¹f ijvt A01 ai b j ðL00 Łij ¦"A11 b1j ¦A33 :R 1 #ðL10 Łij ¦B01 ai b j ðL20 Łij ¦"B11 b1j −A33 :R#ðL30 Łij ¹f ijvh A01 ai b j ðL01 Łij ¦"A11 b1j ¦A33 :R 1 #ðL11 Łij ¦B01 ai b j ðL21 Łij ¦"B11 b1j −A33 :R#ðL31 Łij vs f ijmn −"am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn −A01 bn "am−0 ðL02 Łm−0\n di\m−0 −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn −"A11 bn1 ¦A33 :R 1 #"ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn −B01 bn "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn −"B11 bn1 −A33 :R#"ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724 vf f ijmn −A01 bn "am−0 ðL03 Łm−0\n di\m−0 −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn
−"A11 bn1 ¦A33 :R 1 #"ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn −B01 bn "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn −"B11 bn1 −A33 :R#"ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\n di\m¦0 #d jn ¹f ijvw −
A11 ¦A33 A01 vt B01 vh ¹f ¦ ¹f b j ðLŁij ¦ R R ij R ij
vs vs vf vf f ijvt −f¹ ijvt \ f ijvh −f¹ ijvh \ ¹f ijmn f ijmn \ ¹f ijmn f ijmn \ f ijvw −f¹ ijvw
f wt ij − − f ijwh − − ws ¹f ijmn −
− −
−
¹f wf ijmn − − −
−
A01 A11 ¦A33 a ðL Ł − b j ðL10 Łij R i 00 ij R
0
1
0
1
B01 B11 −A44 ai ðL20 Łij − −A33 b j ðL30 Łij R R
A01 A11 ¦A33 a ðL Ł − b j ðL11 Łij R i 01 ij R
0
1
0
1
B01 B11 −A44 ai ðL21 Łij − −A33 b j ðL31 Łij R R
A01 ðL Ł d −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn "a R m−0 02 m−0\n i\m−0 A11 ¦A33 bn "ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn R
0 0
1 1
B01 −A44 "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn R B11 −A33 bn "ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn R
A01 ðL Ł d −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn "a R m−0 03 m−0\n i\m−0 A11 ¦A33 bn "ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn R
0 0
1 1
B01 −A44 "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn R B11 −A33 bn "ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\n di\m¦0 #d jn R
2722
2723
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0
f ijww − A44 ai1 ¦A33 b1j ¦
A11 R
1
1
ðLŁij −
A01 wt B01 wh f − f R ij R ij
wt ws ws ¹f wt ¹f ijwh f ijwh \ f ijmn ¹ wf ¹f ijww f ijww −f¹ ijmn \ f wf ij f ij \ ijmn −f ijmn \
f ct ij A44 ðL20 Łij f ijch ai ðLŁij ¦A44 ðL21 Łij cs ¹f ijmn A44 "ðL22 Łm−0\n di\m−0 −ðL22 Łm¦0\n di\m¦0 #d jn
¹f cf ijmn ðbn "ðLŁm−0\n di\m−0 −ðLŁm¦0\n di\m¦0 # ¦A44 "ðL23 Łm−0\n di\m−0 −ðL23 Łm¦0\n di\m¦0 #Łd jn f ijcw A44 ai ðLŁij ¦
A01 ct B01 ch f ¦ " f −ai ðLŁij # R ij R ij
ct cs cs ¹f ct ¹f ijch f ijch \ f ijmn ¹ cf ¹f ijcw f ijcw −f¹ ijmn \ f cf ij f ij \ ijmn −f ijmn \
0
¹f ft ij A01 ¦
1
0
¦ B01 ¦
0
¹f ijfh A01 ¦
0
1
B01 B11 1 ai b j ðL00 Łij ¦ A11 ¦ b j ðL10 Łij R R
1
0
1
D01 D11 1 ai b j ðL20 Łij ¦ B11 ¦ b j ðL30 Łij R R
1
0
1
B01 B11 1 ai b j ðL01 Łij ¦ A11 ¦ b j ðL11 Łij R R
0
¦ B01 ¦
1
0
1
D01 D11 1 ai b j ðL21 Łij ¦ B11 ¦ b j ðL31 Łij R R
fs f ijmn −"am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn
0 0 0 0
1 1 1 1
− A01 ¦
B01 bn "am−0 ðL02 Łm−0\n di\m−0 −am¦0 ðL02 Łm¦0\n di\m¦0 #d jn R
− A11 ¦
B11 1 bn "ðL12 Łm−0\n di\m−0 −ðL12 Łm¦0\n di\m¦0 #d jn R
− B01 ¦
D01 bn "am−0 ðL22 Łm−0\n di\m−0 −am¦0 ðL22 Łm¦0\n di\m¦0 #d jn R
− B11 ¦
D11 1 bn "ðL32 Łm−0\n di\m−0 −ðL32 Łm¦0\n di\m¦0 #d jn R
0 fs f ijmn − "am−0 ðLŁm−0\n di\m−0 −am¦0 ðLŁm¦0\n di\m¦0 #d jn R
R[ Mao\ C[!H[ Lu : International Journal of Solids and Structures 25 "0888# 2710Ð2724
0 0 0 0
1 1 1 1
− A01 ¦
B01 bn "am−0 ðL03 Łm−0\n di\m−0 −am¦0 ðL03 Łm¦0\n di\m¦0 #d jn R
− A11 ¦
B11 1 bn "ðL13 Łm−0\n di\m−0 −ðL13 Łm¦0\n di\m¦0 #d jn R
− B01 ¦
D01 bn "am−0 ðL23 Łm−0\n di\m−0 −am¦0 ðL23 Łm¦0\n di\m¦0 #d jn R
− B11 ¦
D11 1 bn "ðL33 Łm−0\n di\m−0 −ðL33 Łm¦0\m di\m¦0 #d jn R
¹f ijfw −
0
2724
1
0 B11 A01 ft B01 fh ¹f ¦ ¹f A11 ¦ b j ðLŁij ¦ R R R ij R ij
fs fs ff ¹ ft f ijfh −f¹ ijfh \ ¹f ijmn f ft f ijmn \ ¹f ff f ijfw −f¹ ijfw ij −f ij \ ijmn f ijmn \
References Flugge\ W[\ 0859[ Stresses in Shells[ Springer!Verlag\ Berlin\ Gottingen\ Heidelberg[ Hui\ D[\ Du\ I[H[Y[\ 0876[ Initial postbuckling behavior of imperfect antisymmetric cross!ply cylindrical shells under torsion[ ASME J[ Appl[ Mech[ 43\ 063Ð079[ Simitses\ G[J[\ Shaw\ D[\ 0874[ Imperfection sensitivity of laminated cylindrical shells in torsion and axial compression[ Composite Structures 3\ 224Ð259[ Stein\ M[\ 0875[ Nonlinear theory for plates and shells including the e}ects of transverse shearing[ AIAA J[ 13\ 0426Ð 0433[ Tabiei\ A[\ Simitses\ G[J[\ 0883[ Buckling of moderately thick\ laminated cylindrical shells under torsion[ AIAA J[ 21\ 528Ð536[