Thin  Walled Structures 15 (1993) 4363
Postbuckling Analysis of Stiffened Cylindrical Shells under Combined External Pressure and Axial Compression
Huishen Shen, a Pin Zhou b & Tieyun Chen b °Department of Civil Engineering, bDepartment of Naval Architects and Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China (Received 7 June 1990: revised version received 4 June 1992; accepted 5 June 1992)
ABSTRACT A new approach is extended to investigate the buckling and postbuckling behaviour of perfect and imperfect, stringer and ring stiffened cylindrical shells of finite length subject to combined loading of external pressure and axial compression. The formulations are based on a boundary layer theory which includes the edge effect in the postbuckling analysis of a thin shell. The analysis uses a singular perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Some interaction curves for perfect and imperfect stiffened cylindrical shells are given and compared well with experimental data. The effects of initial imperfection on the interactive buckling load and postbuckling behaviour of stiffened cylindrical shells have also been discussed.
NOTATION AI, A2 bl, b2 dl, d2 El, E2, E el, e2 F,F
Crosssectional area of stringer and ring stiffener Loadproportional parameter Distance between centers of stringers and rings Elastic modulus of stringer, ring and skin Stringer and ring stiffener eccentricity Dimensional and nondimensional form of stress function 43
44
G1J1, G2J2 Ii, 12 L Hs, nr
Pcr, Plow Poxp, Ppos, R R x, Ry t
W,W W*, W* Z c
Aq, A*
A* # v
Huishen Shen, Pin Zhou, Tieyun Chen
Torsional rigidity of stiffener crosssection Moment of inertia of stiffener crosssection about its centroidal axis Length of a cylindrical shell Number of stringers and rings Theoretical buckling and lowest postbuckling load Experimental buckling and lowest postbuckling load Radius of a cylindrical shell Nondimensional form of stress, defined by R~ = Ap/A ° and Thickness of a cylindrical shell Dimensional and nondimensional form of additional deflection of a cylindrical shell Dimensional and nondimensional form of geometrical imperfection of a cylindrical shell Geometric parameter of a cylindrical shell Endshortening of a cylindrical shell A small perturbation parameter Non dimensional form of load of axial compression Non dimensional form of load of external pressure Theoretical value Of Ap and ,~q for pure axial compression and pure external pressure, respectively Imperfection sensitivity parameter, defined by A* = Ap (imperfect)/Ap (perfect) Imperfection parameter Poisson's ratio INTRODUCTION
In practical engineering structures, stiffened cylindrical shells have found extensive use. A study of the postbuckling behaviour of stiffened cylindrical shells under various loading conditions becomes relevant. Buckling and postbuckling analysis of stiffened cylindrical shells has been performed for some simple loading cases) However, limited progress has been achieved in the methods of buckling analysis of stiffened shells under combined loading. 24 The test which has been carded out on this interactive buckling problem is also limited. 47 Although the classical linear theory is more applicable to closely stiffened shells than to unstiffened ones, the postbuckling loss of stiffness of some stiffened shells can be more severe than of some unstiffened ones. Under combined loading condition, the shape of interaction curves is different between stiffened and unstiffened shells. Until recently, there
Postbuckling analysis of stiffened cylindrical shells
45
was no rule or recommendation for the strength prediction of stiffened cylindrical shells under combined loading. Such a state means that the further theoretical analysis should be done. As described in Refs 1013, a boundary layer phenomenon exists in the buckling of thin shell. A new approach on the boundary layer theory and associated method suggested in Refs 11 and 12 has recently extended to study the postbuckling behaviour of stiffened cylindrical shells under axial compression) 4 The present study is directed specifically to the postbuckling analysis of stiffened cylindrical shells, both stringer and ring stiffened cylindrical shells with and without initial imperfections, subject to combined loading of external pressure and axial compression. In the present analysis, the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometrical imperfection have been put into consideration simultaneously, and the stiffener geometry, the material properties o f stiffeners and skin have been considered as well. The shape of initial geometrical imperfection is assumed as assymetric buckling mode of cylindrical shell. Some numerical examples are presented. These show the effects of initial imperfections on the interactive buckling load and the postbuckling behaviour of stiffened shells. ANALYTICAL FORMULATION A stiffened cylindrical shell with mean radius R, length L and thickness t is considered and for which the geometry is shown in Fig. 1. It is assumed that the shell is subjected to two loads combined out of uniform pressure q and axial load P. Letting nr be the number of ring stiffeners, then L = (nr + 1)d2. The ns stringer stiffeners have spacing dr, eccentricity el n d2 and e2 are defined analogously to dl and el, but are for the ring stiffeners. Generally ns, n,, dl, d2, el and e2 can be different. Denoting the initial deflection by W*, let W be the additional deflections, and F the stress function defined as N x = F,yy,
Ny = F,~,
Nxy = F,xy
then the nonlinear large deflection equations of stiffened cylindrical shells are given as follows  

1
 
L ~ ( W )  La(F)  ~ F,= = L ( W + W*, F ) + q 

1  
1
L2(F) + L3(W) + ~ W,xx =  ~ L(W + 2W*, W)
(1) (2)
46
Huishen Shen, Pin Zhou, Tieyun Chen t d1 L
Fig. !. Geometry and coordinate system of a stiffened cylindrical shell.
where
194 {~4 d4 L, = Dx ~ + 2(Dxv + 219,) ~ + Dv ay4 ( 1 1)0 1 d4 L2  l~v oX4"I 2 ~yxy + ~  ~ ~ 04
04
L3 = f~y ~Zx4 + (f~ + fy) ~ d2 02 L = ax 2 dy 2
4
10 4 + gx   Oy4 04
+ f.vx o /
02 d2 d: 0 2 2 dxdy dxOy + dy 2 dx 2
in these equations, the bending, stretching a n d coupling stiffnesses are included, as defined in Appendix. The corresponding b o u n d a r y conditions are x = O, L;
W = Mx = 0 (simply supported)
(3a)
W = W,x = 0 (clamped)
(3b)
o "e Nxdy + P + a~R 2q = 0
(3c)
where a = 0 a n d a = I for lateral a n d hydrostatic pressure loadings, respectively, a n d we have the closed condition
Postbuckling analysis of stiffened cylindrical shells
foz"Rdy=O OV
47
(4)
The unit endshortening relationship is
L
2~RL
~dy
z,,R L{[ 1 02~
1
F O2W
. O2Wq
1 O2F 1 1/aW\ 2
OW OW*} Ox Ox dxdy
 [Sx
(5)
Introducing nondimensional quantities X = nx/L,
y = y/R,
fl = L/rtR,
e = (n2R/L2)4v/DxD/BxBv, F = e2F/ rl4~" ~
Z = (1  v2)l/2L:/Rt,
(W*, W) = e(W*, W)'v/BxBv/DxD,,,
Dv/~xD.v, v v = BJB~.v, ,
r22 = e.v
"~ ~
Y12 = (Dxy + 2Dk)/Dx, •
7'24 I e.v~ev/ex~ 4
(7'30, 7'32,7'34, 7'3,t, 7'322) = (~v, A + ~ , ~x, fx, ~ ) VBxB~/D~D~,
Ap = P/4s#v/DxDvBxBy , A~ = Pv/3(1  v2)/2,,rEt 2, Aq = q(3)3/4LR3/2/4n(BxBy)t/S(DxDy) 3/8,
A* = q(3)3/2LR 3/2(1  v2)3/4/rrEtS/2v/2 , 6p = (A,
8" = ( A x / L )/(t/R )v/1/3(1  v2),
6q = (Ax/L )(3)3/4LR I/2/4rt(OxOy/BxB.v) 3/8, 6~ = (Ax/L )(3)3/2LR 1/2(1  V2)3/4/TF13/2V/2 enables the nonlinear eqns (1) and (2) to be written in the nondimensional form 62Ll(W)  eY14L3(F)  y14F~ = YI4fl2L(W + W*, F) + Yt4 4 (3)l/4~LqF..3/2
(6) L2(F) + ~ ' 2 4 L 3 ( W ) + 7'24W,xx =  1y24f12L(W + 2W*, W)
(7)
48
Huishen Shen, Pin Zhou, Tieyun Chen
where
LI
=
04 ~xx4Jr" 2 ~,I'I2Pa2 ~ 04
04 + ~/14fl 2 4 ~yy2[
04 '~.~. 02 d4 L2 = fix4 + "r22" ~ 04
L3 = r30 ~
y2 04 04
4 24P O4
O4
+ 2r32fl20xEOy2 + r34fl4
02 02 02 O2 02 02 L = Oxx2 ~  2 OxOy OxOy + Oy2 0 x 2 The boundary conditions become x = O, ;7;
W = M~ = 0 (simply supported)
(8a)
W = W,x = 0 (clamped)
(8b)
1 (2nfl2c)2F
~Jo
2(3)1/4)~q83/2
a;~dY+ 2&~ + a 3
0
(8c)
and the closed condition becomes
fo
.~[ d2F
,2 d2FI
6.,},24[Y3oc)2W "t" ~'322fl 2 ~2y2~ 02 + ~'24W 2
dW dW*~ jay
(9)
= 0
The unit endshortening relationship becomes
[2~3/4 871.2]/24"
+ r34fl 2 a~_~W~
1
/dW'~ 2

OW 3W*~ Y24 Ox ~  ~ d x d y (10)
Applying eqns (6)(10), the postbuckling behaviour of perfect and imperfect, stiffened cylindrical shells under combined loading of axial compression and external pressure can be determined by a singular perturbation technique suggested in Refs 1 1 and 12. Buckling under external pressure alone and buckling under axial compression alone are two special cases of the present interactive buckling problem.
Postbuckling analysis of stiffened cylindrical shells
49
As the small perturbation parameter e < l, for unstiffened cylindrical shell this condition requires that Z > 2.85 (and only this case is meaningful for linear buckling analysis), 13 eqns (6) and (7) are the equations of boundary layer type, from which the nonlinear prebuckling deformation, the large deflection in the postbuclding range and the initial geometrical imperfection together with the effects of stiffener geometry and material properties can be considered simultaneously. To construct an asymptotic solution of the stiffened cylindrical shell, the additional deflection and stress function in eqns (6) and (7) are assumed as
W = w(x, y, e) + l~(x, ~, y, e) + l~(x, (, y, e)
(1 l)
F =f(x,y, e) + if(x, ~,y, e) + F(x, (,y, e) where w(x, y, O,f(x, y, e) are called outer solutions, or regular solutions of the shell, W(x, ~, y, e), F (x, ~, y, e) and 14/(x, (, y, e), F(x, (, y, e) are the boundary layer solutions near the x = 0 andx = n edges, respectively, and ~ and ( are the boundary layer variables, defined as =
xlv ,
(=
 x)lv/
In eqn (1 l) the regular and boundary layer solutions are both taken in the form of perturbation expansions as
w(x, y, e) = Z e'J/2Wj/2(X'y) j=0
(12a)
f(x, y, C,) = Z ~°J/2fj'/2(X'y) j=0
~V(x,~,y,e) =
Z ej/2* 'l~j,,2 + ,(x, ~X,y) j=O
(12b)
(x, ~, y, e) = Z ey/2 +2F~'/2+ 2(x,~, y) j= O
W(x,(,y,e) = Z eJ/2+1~/2 + ,(x, (,y) j=O
F(x,E,y,¢) = Z £J/2+2~'/2 + 2(x,E,y) j=0
and let the buckling load parameters be expanded as
(12c)
Huishen Shen, Pin Zhou, Tieyun Chen
50
4
(13)
2).p~ = g x = E ~°JK':J' 3 (3)1/4'~qg3/2= gv = Z Cjgvj j=O
j=O
The initial geometrical imperfection is assumed to be in the shape of asymmetric buckling mode of the shell
W*(x, y, e) = ~2A~'lsinmxsinny = ~21aA12~sinmxsinny
(14)
in which tt is imperfection parameter and I.t = A "1/A t2), and A t2) = nondimensional amplitude of the 2ndorder term e2AI2)sinmxsinny of wj(x,y) in eqn (12a). As shown in Refs 11 and 12, the effect of boundary layer on the solution of compressive shell is of the order e ~and on the solution of the pressurized shell is of the order e 3/2, thus for cylindrical shell under combined loading two kinds of loading conditions should be considered. Case (1) high values of external pressure combined with relatively low axial load Let P
rtR2q = bl
(15a)
or
2Apt;
_ bl
(15b)
4 (3)V4~qe3n 2 3 in this case, the boundary condition (8c) becomes
1 fz"B2 O2F dy+ (a +
2 (3)1/4~q83/2 = 0
(16)
By replacing (a + bl) with at, and by using a singular perturbation procedure similar to that in Ref. 12, the asymptotic solutions satisfying clamped boundary conditions can be obtained and written as 1 (3 )3/4e3/Z[~(qO)+ Z(q2)(A/2~82)2 + . . . ] Aq = ~
(17)
where I(0) =
Y24m4
1 2)g2 (1+ la)(n2~2+~alm +
1
(2 + ~) ~ ~ [g,
m2
Y2~3
,(n2132+~alm / 1 2I
g2
(1  u 
( 1 ) _~414+1 ( l + u/) 2 (1 + ,u) nZO2 + ~ alto 2
~2
8
51
Postbuckling analysis of
[email protected] cylindrical shells
1
g3
(11,)2
+
m2
&I_ + (4 + [
n2P2
,a,m’
A(2) 4
=
(1 +
1 m4n2B2 ~ 4
y24g:
lo2
&‘,(I
g2
(1 i$PGE
)
(
4P + P2)
Yl4
[ l
1
+
g2
+ Pu)@
+
1 E3
pu) +
(1
; y24n2j?2
+
2ub2
n2p2
+
Jj alm2
Y24n2B2gz
(1 + p)
n2a2 +
ta,m2)g2

h,m”
(
ia,m2(1 [
+ 2j~)
2(1 + /f)2 + n2P2
1 alm2 2 )
+

( +
(1
+
wq2
+
8m4(1
+
P> (2
+‘)]j
g2
and 6,
c!_
;a,y:4

1
b 
“.“)
+
4
1  fWI)EII2j~
t&(
724 [(
+

[ n(3)3'4a
1 1  ~W*
(
r:4
2 E ~2 4 )I
1
1 + 32 (3)3’4 m2( 1 + 2p)Em312+ 2g’3 E“2 (A\‘&‘)’ C
+ ...
(18)
In eqns (17) and (18), (A\21)~2) is taken as the second perturbation parameter relating to the nondimensional maximum deflection, if the maximum deflection is taken at the point (x, y) = (n/2m, n/h), then
and the nondimensional
maximum
deflection
is written as
52
Huishen Shen, Pin Zhou, Tieyun Chen
Case (2), high values of axial compression combined with relatively low pressure.
Let nR2q
= b2
(20a)
or
3 (3)l/4,a.qg3/2
2;%,g
(20b)
 2b2
in this case, the boundary condition (8c) becomes 1 ~2~
Dr
02F
/3 ~
dy + (1 + ab2)2k.pe = 0
(21)
By taking a2 = 2bff(1 + ab2), the asymptotic solutions obtained in Ref. 14 may be extended to stiffened cylindrical shells under combined loading condition. They are 1 m2 k., (1 + ab2) (m 2 (~n23 2) [X~°) A'~2)(At2w%')2+/~'(P4)(A]2?g)4"~'" "] (22) where
1,f y24m2 ei k'~°l = 2 [(1 +//)g2 [g,
+
(2+//) Y2493+ 1 (1 +//)2 g2 (1 +//)m 2
(1////2)y24g32]c + // g3 [ l g3 8] (]+~)2 g2 _I (1 +//)~ m 4 (1 +//)m 2 J
[g, (4 + / / + 112) g24g32]t2~ ~'~ (1+//)2 ~ J J l.j ]_( _y2 + 1( T~4 )t{:2)= 8[2\?'t4g24 + ?'24) ?'2'm6(2 //) cI g~  2\?',47"~+ ?'2,) m 4 [r34 (1 +//)2 + (1 + 2//) +//(3 +//) r~a3] g2 L~24 _
1__(
(1+//)
(1+/1)
g2 _1
Y,4 .2 )m2( 1 + 2//)g + (y24m2n'e4~
4\?'147'24 I 1'34]
\
tm2 )
g2
J
]
g2[(4 + 9//+ @2) + m 2 4 a2n2/~ (1 + 2~) + 8m40 +//)(2 +//) m2 &(1 +//)  4m4(m2\ 4a2n2/~2 )
e
53
Postbuckling analysis of stiffened cylindrical shells
1_(
_
T]4_ ~ mZg3
4\y14Y24+ Y~4) g2 [Y34~24(4 + 1241(1++15//2//)2+ 4//3) + 2(4 + (1//++2//2//)2+//3) z24gale~g2 _! J
/~(p4,= 1 ( }#24 ) 2y24m'°(1 +//) ei 128 \~/14]:24"IL'}/24 g3 g ,~[k~_ [ {m + 9a2n2f12~ ~) (I + 3//+//2) fm 2 + 5a2n2~2)
+ ~,m~5ra2n~) (4 + 2//) + ( 1 +//)
]
]}
+ ~/(6 +g2 r(m [~ m2%
+ 8//+ 2uz)  (2//+ 3//3 +//3) [
[ :m2+9a2n2B2'
//)]
g'3k m2 +
+
and 6p = (1 + +
ab2)[(Y24

a2) el/21Xpj a2O.v)  n4_ ba_ o~(Ov.~_Y24
[(l + ab2)2 (°v2~24a2)2b e"/2]
+
  e 1:2 e +
e 3 (Al~)e) 2
na
1 {m2n4/34(1+//)2
V g2(1"1.... 2]1) + 8m4(l_m7+//) ] 2E3} Lg2(1 + / / )  4m4(m2T ~  ~ 2)
(23) in eqns (22) and (23), similarly, (AI])e)is taken as the second perturbation parameter in this case, and we have (.4t]~) ' + . . .
tkYi4Y24 + Y~  32\YI4Y24 + y324~ + 2(%

Y24
a2) (
mz
16n202g2 g2 a
LY24
'~a(2)].W2
\m z 4~n202} "'" J
"" " + " "
(24a)
54
Huishen Shen, Pin Zhou, Tieyun Chen
and the nondimensional m a x i m u m deflection is written as
Wm= Wm44Bxnv/OxOv_
. +
2(0"v]/24a2) (m2 +m2a2n2f12]~AI°)p
g2, g3 and gt = m 4 + 2y12m2n2fl2 + ~/24//4~4 g2 = m 4 + 2y22m2n2fl2 + ~24H4~4
In all above equations gl,
(24b)
g13 are defined as
g3 = O30m4 + y32m2n2fl 2 + Y34n4fl 4 g13 = m4 + 18y22m2t12fl 2 + 81y24r/4fl 4
Equations (17) and (22) characterize the postbuckling loaddeflection curves of stiffened cylindrical shells under combined loading. By increasing bt and b2, respectively, the interaction curve of a stiffened cylindrical shell under combined loading can be constructed with these two lines. Note that since b2 = 1/b~, only one loadproportional parameter should be determined by experiment. In eqn (17), ifb~ = 0, or a~ = a, we obtain the postbuckling loaddeflection curve of stiffened cylindrical shells under external pressure alone, and in eqn (22), if b2 = 0, or a2 = 0, we obtain the postbuckling loaddeflection curve of stiffened cylindrical shells under axial compression alone. The buckling load of perfect shell can readily be obtained numerically, by setting___~= 0 (or */t = 0), while taking the m a x i m u m deflection W,, = 0 (or W/t = 0). It should be pointed out from that, due to the effect of the boundary layer, the nonlinear prebuckling deformation is taken into account, thus the result presented is different from the classical one. As described in Ref. 14, when the shell is long enough, the small perturbation parameter e approaches zero. Only in this case may the effects of boundary layer be completely neglected. Indeed, the results calculated indicate that the effect of boundary layer is small as shell geometric parameter Z > 1000, and may be neglected in design analysis.
N U M E R I C A L RESULTS AND DISCUSSION The buckling and postbuckling behaviour of a stiffened cylindrical shell was investigated analytically using a program developed for the purpose and many examples have been solved numerically, including the following. The initial buckling loads and the lowest postbuckling loads of some
Postbuckling analysis of stiffened cylindrical shells
55
outside stringer stiffened cylindrical shells are calculated using data from Ref. 9. The results obtained are listed in Table 1 and compared with the experimental and numerical results given by Singer and Abramovich.9 The boundary condition for experimental shell is realistic, and is close to clamped edge condition. In Table 1, Pexpand Ppost are experimental buckling and lowest postbuckling loads; Per and Proware theoretical buckling and lowest postbuckling loads. Clearly, the results obtained from present method are in reasonable agreement with the experimental results and are better than theoretical results from Ref. 9 when a small imperfection is taken into account. The postbuckling loaddeflection curves of outside ringstiffened cylindrical shells under pure lateral pressure, which failed by the general instability mode, are shown in Fig. 2 and compared with the experimental data given by Seleim and Roorda. 8 It can be seen that the postbuckling equilibrium path of pressurized modelate length stiffened cylindrical shell slopes up gently, such a state means that the postbuckling behaviour is stable. As small imperfection is taken into account, the agreement between theory and experiment is reasonable. Figure 3 shows the interaction curves of perfect and imperfect stringer stiffened cylindrical shells under combined hydrostatic pressure and axial load. The experimental results are from Abramovich et al. 6 The interaction curve is contructed with two lines both m = 1, the transition from one to another is smooth, and they appear as one line. An imperfection causes the reduction of the buckling loads of stringer stiffened cylindrical shells when the axial load is a major factor. Figure 4 shows the interaction curves of ring stiffened cylindrical shells under combined lateral pressure and axial load. The experimental results are from Midgley and Johnson. 7The shape of interaction curve of ring stiffened cylindrical shells differs from that of stringer stiffened ones. The interaction curve is constructed with two lines ofm = 19 and m = 1, and the interaction of these two lines occurs. The buckling mode will be changed as load proportional parameter b~ increases. The good correlation between the present results and referenced results of Table 1 and Figs 24 reveals the good accuracy of the method presented. The curves of imperfection sensitivity of stringer stiffened cylindrical shells under combined loading condition case (2) are shown in Fig. 5. It may be seen that, similar to unstiffened cylindrical shells, for most practical values of Z, the imperfection sensitivity decreases as geometric parameter Z increases and the loadproportional parameter b2 increases. In this case the stringer stiffened cylindrical shell is imperfection
TABLE I
85
85
85
85
85
85
II0'0 120'1 0'257
110"0 120"1 0'253
154'0 120'1 0"253
154'0 120'1 0253
130'0 120"1 0"252
130"0 120"1 0"254
0"59
0"68
0"70
0"70
0'70
0"70
At dtt
It dlt ~
GtJi diD
Pi~l, (kg)
Pl ..... (kg)
2'83
8'3864 4900(I,9)
2'75
8'2359 3615(I,8)
342
1"68 6"5632 4025 (I, 9)
3 " 8 7 2"59 6"0468 3580 (1,9)
3"94
3 " 9 5 2"78 8"2643 3400(1,8)
3'97
2250
3000
3038
2970
4040
1"09
0"93
0'91
0'86
1"01
0'94
SS3 h
062
0"51
0"50
0"47
0"55
0"51
C4"
Pevl,//Pl .....
Singer & Abramovich t°
3 " 9 5 2 " 8 0 8"1642 4700(1,9)* 4150
el t
"Values within the parentheses indicate the buckling mode (m, n). hLoaded edges assumed simply supported. 'Loaded edges assumed clamped.
n,
L R t (ram) (ram) (ram)
0"83 1"00
00 4338(!,10) 0"13 3591(I,10)
1'03
0"86 0"99
4200(I,9) 3647(I,9)
00 0"10
3918 (I, I0)
0"81 0'97
4208(I.9) 3491(1,9)
00 0"15
0"0
0"93 1.00
0"0 5274(I,10) 0"03 4897(I,10)
e,,,,,/e,r
0'87 0.99
Per (kg)
5433(1,10) 4750(1,10)
00 0"07
W all
2783
3316 3117
3031 2916
3043 2871
4579 4490
4716 4502
p~o. (kg)
Present
081
0"90 0'96
I'00 1"04
0'98 1"03
0"88 0.90
0'88 092
p~,,Jeh,.
Buckling Loads and the Lowest Postbuckling Loads of Outside Stringer Stiffened Cylindrical Shells (E = 7500 kg/mm', v = 0"3)
~.
~. N
"7"
tun
Postbuckling analysis of stiffened cylindrical shells 2 I]
x Experiment (9) Model 3 (nr=13)
~t1_[0 ~" " t O ' 0 1
I}

(a) 1
x Experiment (9) Model5 (n r=11)
I
b Cr
2
~/t
3
xExperiment (g) Model 8 (nr=9)
57
"l
~'. I o T °'°I
ol.
 
(b) 1
2 lj I
1 2
3
xExperiment (9) Model 10 (nr=13)
1
w/t
~'I (7
'
:I
0
_Wt
{g
I
:I
'01    
=
I
(°'3
I
1
2
0
~,_~ o ~
" 10.01
[

I
'
I
1
2
3
Fig. 2. Postbuckling loaddeflection curves of ring stiffened cylinders under lateral pressure.
1.0
,
stiffened •Stringer DUD8
1"01~ ~.\
Model
" \
RIt =482
,276
Stringer stiffened Model DUDIO
.,;:,,~,
I\\
x 05 oc
~,.f
o
¥ " tO'04 
05 Ry (a)
\
J ~*~o
~
I ~ "(0"05
1.0
0.5 Ry
\\
"~
10
(b)
Fig. 3. Interaction buckling curves of stringer stiffened cylinders under combined hydrostatic pressure and axial load.
58
Huishen Shen, Pin Zhou, Tieyun Chen
1.o
m=19 \ ,, Ring stiffened R/t = 267 z = 1435 \ ~ . ~ Model A  1 / ~
Model A  2 f Model A3 ~
a:~ 0"5
m=l
_~\1 \\
o Experiment ( 7 ) ~
~
I
05 Ry
10
Fig. 4. Interaction buckling curves of ring stiffened cylinders under combined lateral pressure and axial load.
Stringer stiffened Outside 1"0 
L
Stringer stiffened 1.0 :
Inside
Outside Inside

i
b2=O 0.5
RIt=500
b2=O.O5
Aj
I1
dlt 0.5 O
_el
t ~ .*3.0 1.83 I
0.5
0.5
RIt=500
G_121 dlD 100
dlt (a)
1O
os
z=1OO0
el
I1
t
~3
..3.0 I.a3 =,,
I
0"5
G!J1 40 1o.o
(b)
1"0
~tt (a)
(b)
Fig. 5. Comparisons of imperfection sensitivities of stringer stiffened cylinders under combined loading.
sensitive, and an initial imperfection causes the reduction in the buckling load. Generally, the outside stringer stiffened shell is more sensitive than the inside one. It is noted that the effect of initial imperfection on stringer stiffened
Postbuckling analysis of stiffened cylindrical shdls
59
cylindrical shells is insensitive under loading condition case (1), except for short cylinders. Thus in this case the experimental data follow the theoretical interaction curve reasonably well. Figure 6 present typical postbuckling loadshortening curves and loaddeflection curves for perfect and imperfect stringer stiffened cylindrical shells under combined loading. It shows that the buckling loads as well as the lowest postbuckling loads decrease as load effects increase (load proportional parameter b2 increases), and the postbuckling equilibrium paths tend to be gently smoothed out as well. Next, it is evident that the buckling loads and the lowest postbuckling loads are reduced when initial imperfections are taken into account, and the effect of initial imperfection on the buckling loads is stronger than that on the lowest postbuckling loads. Figure 7 present typical postbuckling loadshortening curves and loaddeflection curves of ring stiffened cylindrical shells under combined loading with asymmetrical buckling mode. It is found that the effect of
Stringer stiffened (outside)
RIt =500 z : 500
I
Stringer stiffened (outside)
RIt =500
Z=500
b2=O(1,10) QCL
=0.02 (1,10)
r<
= (a)
0.5
1.0
ol
o.~~.o
S t r i n g e r stiffened ( i n s i d e ) R/t =500 z:500
,.,
,~.o
2
1.5
(b) I
4
6
S t r i n g e r stiffened ( i n s i d e )
A1
el 11 G1J1 t ~'t3 dl D 0.,5 *3.0 1.83 10.0 dlt
T "{O's~b2:0 (1,~o)
~*_ o 1
~/
b2=0.02 (I,11)
~ /~" ~
/ b2= 0 (1,10)
b
2
=
0
~
2
(1,11]
i
I
0"5
1.0
(d)
Fig. 6. Postbuckling equilibrium paths of stringer stiffened cylinders under combined loading.
60
Huishen Shen, Pin Zhou, Tieyun Chen Ring stiffened (inside) R/t = 267 z = 1437
Ring stiffened ( inside ) RIt 267 z =1435 b2=O (19,8)
b2=O (19,8)
~ b 2 = 0.15 qlcI.
~ 2 = 0 . 1 5
tlO.
J
//f~f
(19,8)
1
~* ,f° T  LO.~
,//~A2
e2
I_.22
/
"T"
d2t3
(a) ;
I
I
1
5"~ 2
3
~
0.36752'25
0"3752
O2J2 d2D 2.666 (b)
I
I
1_ Wit
2
Fig. 7. Postbuckling equilibrium paths of ring stiffened cylinders under combined loading. load proportion on the buckling loads of ring stiffened cylindrical shells is weaker than that of stringer stiffened ones, and the equilibrium path of ring stiffened cylindrical shells resembles the corresponding curves of fiat plates, and is stable as usual. In general, the effect of initial imperfection on the postbuckling equilibrium path of ring stiffened cylindrical shells is insensitive.
CONCLUSIONS Postbuckling analysis and an associated analyticalnumerical procedure have been presented for stiffened cylindrical shells. A number of examples have been given to illustrate their applications, which relate to the performance of perfect and imperfect, stringer and ring stiffened cylindrical shells. Numerical correlation with existing test data are reasonably good. In relation to stiffened cylindrical shells subjected to external pressure and axial compression, the following conclusions can be drawn from the results obtained. Evident differences are existed in the interaction curves of stringer and ring stiffened cylindrical shells under combined loading condition. The postbuckling behaviour of stringer stiffened cylindrical shells is unstable and imperfectionsensitive. The imperfection sensitivity decrease as load effects increase. In contrast, the postbuckling behaviour of a ring stiffened cylinder is complicated by the many and varied forms that it might take. In general, the postbuckling behaviour of ring stiffened cylinders is stable and imperfectioninsensitive.
Postbucklinganalysis of stiffened cylindricalshells
61
The postbuckling b e h a v i o u r of interring cylinder is similar to that o f unstiffened cylinder which m a y be seen as a special case of the present problem 13.
REFERENCES 1. Esslinger, N. & Geier, B., Postbuckling Behavior of Structures. SpringerVerlag, Berlin, 1975. 2. Simitses, G. J., Buckling of eccentrically stiffened cylinders under combined loads. A/AA J.. 7 (1969) 3357. 3. Cross, J. G. A., Stiffened cylindrical shells under axial and pressure loading. In Shell Structures."Stability and Strength, ed. R. Narayanan. Elsevier Applied Science Publishers, Barking, 1985, pp. 1956. 4. Tennyson, R. C., The effect of shape imperfection and stiffening on the buckling of circular cylinders. In Buckling of Structures. ed. B. Budiansky. SpringerVerlag, Berlin, 1976, pp. 25173. 5. Walker, A. C., Segal, Y. & McCall, S., The buckling of thinwalled ringstiffened steel shells. In Buckling of Shells, ed. E. Ramm. SpringerVedag, Bedim 1982, pp. 275304. 6. Abramovich, H., Weller, T. & Singer, J., Effect of sequence of combined loading on buckling of stiffened shells. Exp. Mech.. 28 (1988) 113. 7. Midgley, W. R. & Johnson, A. E., Experimental buckling of internal integral ringstiffened cylinders. Exp. Mech., 7 (1967) 14554. 8. Seleim, S. S. & Roorda, J., Theoretical and experimental results on the postbuckling of ringstiffened cylinders. Mech. Struct. & Mach., 15 (1987) 6987. 9. Singer, J. & Abramovich, H., Vibration techniques for definition of practical boundary conditions in stiffened shells. A/AA J., 17 (1979) 7629. 10. Scheidl, R. & Troger, H., A comparison of the postbuckling behavior of plates and shells. Computer & Structures. 27 (1987) 15763. 11. Shen, H. S. & Shen, T. Y., A boundary layer theory for the buckling of thin cylindrical shells under axial compression. In Advances of Applied Mathematics and Mechanics in China, Vol. 2, ed. W. Z. Chien & Z. Z. Fu. Int. Academic Publishers, 1990, pp. 15572. 12. Shen, H. S. & Shen, T. Y., A boundary layer theory for the buckling of thin cylindrical shells under external pressure. Appl. Math. Mech., 9 (1988) 55771. 13. Shen, H. S. & Chen, T. Y., Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression. ThinWalled Structures, 12 (1991) 32134. 14. Shen, H. S., Zhou, P. &Chen, T. Y., Buckling and postbuckling of stiffened cylindrical shells under axial compression. Appl. Math. Mech., 12 (1991) 1195207.
62
Huishen Shen, Pin Zhou, Tieyun Chen
APPENDIX The bending, stretching and coupling stiffnesses in eqns (1) and (2) are defined as D.,. = Et 3 12(1  02) + E dlt 3 + E
I1 +ff~t(1o ~,A2
2)
}(~ ~,e,)2v2> dlt t Et ~ At
~ ~ff(1 [ Dv=Et3
02)]
_
02
l E2 I2 Et At (et] 2 12(1 05) + E d2t 3 +ff ~ 2 t \ t ]
1 +El Ai
E dlt (1  02)
[,+~,~ ~,,(l ~,  o2q[,+
A2e2 "(1 0 2) d2t t Et  ~At 02)] _ o5
,,_
1 D,:,. = vEt 3 12(1  02)
(~ A,e_,ye_2 A, e2) +
dlt t Jk E d2t t ] ( 1  ° 2)
A2 _ 02)] _ [I+E)E ~lt( A, 1 _ 02)] [1 + E2 E ~2t(1
Dk D[ =~
1 B~
Et
1
1
Bv
1
l fGidi
(lu)+~\d)D
02]
G2J2]I +~2D) J
Et A2 02)] (1 1 + ~ ~2t(1 
02)
[,+.,~ ~lt A, ( l  o2,][1 + ~2A2 °2>]  v2 ~ ~2t ( 1 [ I + E' ~~L~t (1 o2)l (1 u2)
Et [1 +E, A1 _ o 2 ) ] [ 1 E, A2 02)] 02 ~ ~lt(l + E ~2t(1
Postbucklinganalysis of stiffened cylindrical shells o
1
~.v
(1  02)
Et [ I + E~' ~(1 A'
 02)] [l+ E2  ~ A2 (1 02)]  02
[l+E2E d2t A2 (1 _oZ)] (~_2 A, e, dd t ) (1  02) L=t [l+ E' A'  o2)][1+ E2 ~ ~(1 ~  ~A2 (1 02)]  02 d2t
E ~:t (1 
L=t 1
+ Et ~_ ~t(1 _ 02)] [1 + E2 ~ ~A2 (1
02)]  oz
(~ A i el
i ) (1  o 2)
f~v = ot E
f.vx = or
~lt( 1 _
E2 A2
o2)] o2
d2t
[1 + E, A1 1 _ o2)][l+_~_.~d(l_ E ~lt( Ez A2 o2)] _ o2
63