Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 3, pp. 641-653, Warsaw 2014

BUCKLING OF CYLINDRICAL SHELLS UNDER EXTERNAL PRESSURE

IN A HAMILTONIAN SYSTEM

Jiabin Sun

State Key Laboratory of Structural Analysis for Industrial Equipment and School of Ocean Science and Technology,

Dalian University of Technology, Panjin, P.R. China; e-mail: jbsun1983@dlut.edu.cn

Xinsheng Xu

State Key Laboratory of Structural Analysis for Industrial Equipment and Department of Engineering Mechanics,

Dalian University of Technology, Dalian, P.R. China

C.W. Lim

Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, P.R. China

In this article, the elastic buckling behavior of cylindrical shells under external pressure
is studied by using a symplectic method. Based on Donnell’s shell theory, the governing
equations which are expressed in stress function and radial displacement are re-arranged
into theHamiltonian canonical equations.The critical loads andbucklingmodes are reduced
to solving for symplectic eigenvalues and eigenvectors. The buckling solutions are mainly
grouped into four categories according to the natures of the buckling modes. The effects
of geometrical parameters and boundary conditions on the buckling loads and modes are
examined in detail.

Keywords: buckling, cylindrical shells, external pressure, Hamiltonian system

1. Introduction

With the development of the ocean science and underwater project, the safety criterion for
predicting the collapse of the pressurized vessels and pipelines is necessary in the engineering
application. In the past, this kind of problemwas investigated theoretically and experimentally
by many researchers (Mises, 1914; Flügge, 1973; Batdorf, 1947; Nash, 1954; Galletly and Bart,
1956, Armenakas and Herrmann, 1963; Soong, 1967). In these studies, only a simple one-term
mode functionwas adopted, and theproblemwas solvedunder some special boundaryconditions
which need to be satisfied by an assumed expression. By employing the similarmethod, Yamaki
(1969) and Vodenitcharova and Ansourian (1996) carried out more extensive research and pre-
sented some thought-provoking solutions for this problem under various boundary conditions.
Meanwhile, for the sake of resolvingmore practical problems, numerous approximate and nume-
rical methods have sprung up and exhibited an excellent performance in handling complicated
situations. With the aid of the Ritz method, Tian et al. (1999) investigated elastic buckling of
cylindrical shells with ring-stiffeners under pressure. It was appropriate for any combination of
end conditions by using polynomial functionsmultiplied by boundary equations raised to appro-
priate powers as theRitz functions. Pinna andRonalds (2000)] examined eigenvalue buckling of
cylindrical shells subjected to hydrostatic load under various boundary conditions through an
energymethod. The effect of ends conditions, including radial elastic restraint at the open end,
was discussed in detail. Xue and Fatt (2002) obtained analytical solutions for elastic buckling
of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure. The finite
element method was applied to examine the validity of analytical method and the results were
found to be in close agreement with the numerical method (Goncalves et al., 2008). A set of



642 J. Sun et al.

experimental tests were also conducted by Hübner et al. (2007) to improve the assessment pro-
cedure for cylindrical shells. For the post-buckling analysis of cylindrical shells under external
pressure, the numerical results obtained by means of the nonlinear finite element method were
compared with the results of the experimental study (Aghajari et al., 2006). In order to trace
the nonlinear equilibrium paths, the “Arc-Length-Type Method” was used in the study. Shen
(2008) also developed a boundary layer theory for the similar problem and applied the pertur-
bation technique to determine the buckling pressure and post-buckling equilibrium paths.More
comprehensive results and discussions can also be found in themonographs byTeng andRotter
(2004), Ventsel and Krauthammer (2001).

However, most of the traditional analytic methodsmentioned above belong to the Lagrange
solving system. It involves only one set of variables and can be resolved by the force method
or the displacement method. In this system, the fundamental equations exist in form of high-
order partial differential equations which are difficult to be analytically or numerically worked
out. Recently, Zhong (2004) developed a symplectic analytical method for some fundamental
problems in solid mechanics. Through the Legendre transformation, Lagrange formulations can
be transformed into Hamiltonian dual equations by introducing dual variables. By employing
separationof thevariables, the fundamentalproblemcanbeboileddownto solving for symplectic
eigenvalues and eigenvectors. According to the completeness theorem of the symplectic system,
all solutions can be sought out for the current problem. The symplectic solving approach is
rigorous and rational in solving the problem, and boundary conditions are satisfied in a natural
manner. Xu et al. (2006) investigated the local buckling and the propagation (and reflection)
of axial stress waves by introducing a Hamiltonian system or a symplectic system into dynamic
buckling of cylindrical shells.

In this study, a newHamiltonian system is established to investigate buckling of cylindrical
shells under external pressure. Hamiltonian canonical equations are derived from the Hamilto-
nian principle ofmixed energy. According to rational deduction, buckling characteristic parame-
ters should be determined by solving for eigenequations in a symplectic space.Auniform solving
process is developed for this problemunder symmetric andnon-symmetric boundary conditions.
The factors which influence buckling results are also discussed in detail.

2. Basic equations

Consider a thin-walled cylindrical shellwith radius R, length l, thickness h, Young’smodulus E
and Poisson’s ratio ν (Fig. 1), compressed by uniform lateral pressure P. A circular cylindrical
coordinatewithan x-axis along thecentral axis is adopted.Andthe correspondingdisplacements
can be denoted that x-direction is u, θ-direction is v, z-direction is w, respectively. The
membrane internal forces are given by

Nx =K(εx+νεθ) Nθ =K(εθ +νεx) Nxθ =
1

2
Kεxθ(1−ν)

Mx =D(κx+νκθ) Mθ =D(κθ +νκx) Mxθ =D(1−ν)κxθ
(2.1)

where D = Eh3/[12(1− ν2)] and K = Eh/(1− ν2). {Nx,Nθ,Nxθ} and {Mx,Mθ,Mxθ} are
the resultant membrane forces and bendingmoments. The strain components {εx,εθ,εxθ} and
curvature components {κx,κθ,κxθ} are expressed by



Buckling of cylindrical shells under external pressure in a Hamiltonian system 643

εx =
∂u

∂x
+
1

2

(
∂w
∂x

)2
κx =−

∂2w

∂x2

εθ =
1

R

(∂v
∂θ
−w
)
+
1

2R2

(∂w
∂θ

)2
κθ =−

1

R2
∂2w

∂θ2

εxθ =
1

R

∂u

∂θ
+
∂v

∂x
+
1

R

∂w

∂x

∂w

∂θ
κxθ =−

1

R

∂2w

∂x∂θ

(2.2)

By introducing a stress function F, the internal forces can be written into

Nx =
1

R2
∂2F

∂θ2
Nθ =

∂2F

∂x2
Nxθ =−

1

R

∂2F

∂x∂θ
(2.3)

Fig. 1. Geometric parameters of a cylindrical shell under pressure

Assuming the membrane pre-buckling state, the pre-buckling internal forces are Ñx = 0,
Ñθ =−PR, Ñxθ =0. Total potential energy, caused by the incremental buckling displacements
(u,v,w) and the stress function F, consists of the extension potential energy, bending potential
energy and the potential of external forces. Neglecting higher-order nonlinear terms, it can be
obtained as

Π =Πε+Πk−Πw

=

∫∫

S

[ 1
R2
∂2F

∂θ2
∂u

∂x
+
∂2F

∂x2
1

R

(∂v
∂θ
−w
)
− 1
R

∂2F

∂x∂θ

(1
R

∂u

∂θ
+
∂v

∂x

)]

−
1

2Eh

{(∂2F
∂x2
+
1

R2
∂2F

∂θ2

)2
−2(1+ν)

[∂2F
∂x2
1

R2
∂2F

∂θ2
−
(1
R

∂2F

∂x∂θ

)2]}

+
1

2
D
{(∂2w
∂x2
+
1

R2
∂2w

∂θ2

)2
−2(1−ν)

[∂2w
∂x2
1

R2
∂2w

∂θ2
−
(1
R

∂2w

∂x∂θ

)2]}

− Ñθ
2R2

(∂w
∂θ

)2
Rdx dθ

(2.4)

A Lagrange density function can be derived from Eq. (2.4) as

L=
1

R2
∂2F

∂θ2
∂u

∂x
+
∂2F

∂x2
1

R

(∂v
∂θ
−w
)
−
1

R

∂2F

∂x∂θ

(1
R

∂u

∂θ
+
∂v

∂x

)

−
1

2Eh

(∂2F
∂x2
+
1

R2
∂2F

∂θ2

)2
+
1

2
D
(∂2w
∂x2
+
1

R2
∂2w

∂θ2

)2
−
Ñθ
2R2

(∂w
∂θ

)2
(2.5)

Based on the Hamiltonian principle, the variational equation is expressed as

δΠ = δ

2π∫

0

Rdθ

l/2∫

−l/2

L(F,w) dx=0 (2.6)



644 J. Sun et al.

Then Eq. (2.5) is substituted into Eq. (2.6) and the variation with respect to F and w, is
respectively taken. The compatibility condition and the equilibrium equation are obtained as

δΠ

δF
=
1

Eh
∇4F + 1

R

∂2w

∂x2
=0

δΠ

δw
=D∇4w− 1

R

∂2F

∂x2
+
Ñθ
R2
∂2w

∂θ2
=0 (2.7)

where ∇2 = ∂2/∂x2+∂2/(R2∂θ2) is the Laplacian operator.

3. Symplectic system

The dimensionless terms X = x/R, W/R, φ = F/(Eh3), L = l/R, H = h/R, β = αH2

and Ncr = Ñθ/D are adopted. The dimensionless critical pressure Ncr relates physical pa-
rameters with geometric parameters. An over-dot represents differentiation with respect to θ,
namely Ẇ = ∂W/∂θ in which the θ-coordinate is chosen as a time-equivalent coordinate and,
∂XW = ∂W/∂X in which X-coordinate is taken as a transverse coordinate.
Define twonewvariables ξ=−Ẇ and ϕ=−φ̇.ThedimensionlessLagrangedensity function

is expressed as

L=−αW∂2Xφ−
1

2
β(∂2Xφ+ φ̈)

2+
1

2
(∂2XW +Ẅ)

2−
1

2
Ncr(Ẇ)

2 (3.1)

Define a vector q= {W,ξ,φ,ϕ}T = {q1,q2,q3,q4}T, the dual vector p can be deducted as

p1 =
δL

δq̇1
=−(

...
W +∂

2
XẆ)−NcrẆ p2 =

δL

δq̇2
=−(Ẅ +∂2XW)

p3 =
δL

δq̇3
=β(φ+∂2Xφ̇) p4 =

δL

δq̇4
=β(φ̈+∂2Xφ)

(3.2)

The dual variables denote the equivalent transverse shear stress, bendingmoment, in-plane
shear stress and normal stress, respectively. Substituting the dual variables into Eq. (3.1), the
Hamiltonian density function can be obtained as

H(q,p) =pTq̇(q,p)−L(q,p)

=−p1q2+
1

2
p22+p2∂

2
Xq1−p3q4−

1

2β
p24+p4∂

2
Xq3+αq1∂

2
Xq3+

1

2
Ncr(q2)

2
(3.3)

Substituting Eq. (3.3) into Eq. (2.6), we have

δ

∫
[pTq̇(q,p)−H(q,p)] ds=0 (3.4)

Then the Hamiltonian canonical equations are obtained by integration by parts as

q̇=
δH

δp
ṗ=−δH

δq
(3.5)

Equations (3.5) can be expressed in the matrix form as
{
q̇

ṗ

}
=

[
A B

C −AT
]{
q

p

}
(3.6)

where

A=




0 −1 0 0
∂2X 0 0 0
0 0 0 −1
0 0 ∂2X 0




B=




0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 −1

β






Buckling of cylindrical shells under external pressure in a Hamiltonian system 645

C=





0 0 −α∂2X 0
0 −Ncr 0 0
−α∂2X 0 0 0
0 0 0 0





Define a state vector ψ = {qT,pT}T, then Eq. (3.6) can be simplified as

ψ̇ =Hψ (3.7)

To discuss the property of the matrix H, an inner product needs to be defined as

〈ψ1,ψ2〉=
L/2∫

−L/2

(qT1p2−qT2p1) dX (3.8)

It can be proved that H is a Hamiltonian operator matrix (Zhong, 2004).

4. Symplectic eigensolutions and orthogonality relation

In a symplectic system, by separation of variables, the solution of Eq. (3.7) is expressed as

ψ(X,θ)=η(X)eµθ (4.1)

where η = {q1,q2,q3,q4,p1,p2,p3,p4} and µ are the eigenvector and the eigenvalue. Substitute
Eq. (4.1) into Eq. (3.7), the eigenvalue equation can be obtained as

Hη(X)=µη(X) (4.2)

According to the property of revolutionary shell, Eq. (4.1) needs to satisfy the closed condi-
tion

ψ(X,0)=ψ(X,2π) (4.3)

So it is proved that µn =ni (n=0,±1,±2, . . .). Substituting the corresponding eigenvalues
µn =ni into Eq. (4.2), the characteristic polynomial can be written as

λ8+aλ6+ bλ4+ cλ2+d=0 (4.4)

where a=−4n2, b=6n4−n2Ncr+α2/β, c=2n4Ncr−4n6 and d=n8−n6Ncr. If n 6=0, the
eigensolutions can be classified into four sorts. They can be given by:

Sort 1: If λ1, λ2, λ3 and λ4 are complex roots, besides, λ1 = λ3, λ2 = λ4, |λ1| 6= |λ2|, αi and
βi are absolute values of the real and imaginary part of the characteristic root λi, respectively,
λi =−λi+4 (i=1,2,3,4). It is expressed by

ηn = c1e
α1X cos(β1X)+c2e

α1X sin(β1X)+c3e
−α1X cos(β1X)+c4e

−α1X sin(β1X)

+c5e
α2X cos(β2X)+c6e

α2X sin(β2X)+c7e
−α2X cos(β2X)+c8e

−α2X sin(β2X)
(4.5)

Sort 2: If λ1, λ2, λ3, λ4 are different real roots, λ5, λ6 are complex roots, besides, λ5 = λ6,
α1 and β1 are absolute values of the real and imaginary part of the characteristic root λ5,
respectively, λi =−λi+2 (i=5,6). It is given by

ηn = c1e
λ1X +c2e

λ2X +c3e
λ3X +c2e

λ4X +c5e
α1X cos(β1X)+c6e

α1X sin(β1X)

+c7e
−α1X cos(β1X)+c8e

−α1X sin(β1X)
(4.6)



646 J. Sun et al.

Sort 3: If λ1, λ2 are conjugate pure imaginary roots, λ3, λ4 are different real roots, λ5, λ6 are
complex roots, besides, λ5 =λ6, β1 are absolute value of λ1, α2 and β2 are absolute values of
the real and imaginary part of the characteristic root λ5, respectively, λi =−λi+2 (i=5,6). It
is that

ηn = c1cos(β1X)+c2 sin(β1X)+c3e
λ3X +c4e

λ4X +c5e
α2X cos(β2X)

+c6e
α2X sin(β2X)+c7e

−α2X cos(β2X)+c8e
−α2X sin(β2X)

(4.7)

Sort 4: If λ1,λ2 are conjugate pure imaginary roots, λ3,λ4, . . . ,λ8 aredifferent real roots, β1 are
absolute values of λ1. It is expressed by

ηn = c1cos(β1X)+c2 sin(β1X)+c3e
λ3X+c4e

λ4X+c5e
λ5X+c6e

λ6X+c7e
λ7X+c8e

λ8X (4.8)

where ck = {c1k,c2k, . . . ,c8k}T (k=1,2, . . . ,8) are eight constant vectorswhich canbedetermined
from boundary conditions.
For a special case n= 0, λ= 0 is a quadruple root of Eq. (4.4). So the eigenvector can be

written into

η0 = c1e
λ1X +c2e

λ2X +c3e
λ3X +c4e

λ4X +c5X
3+c6X

2+c7X+c8 (4.9)

By considering the specified boundary condition, it can be proved that this equation have only
a trivial solution for incremental components. So, there are no axisymmetric buckling modes
for buckling of cylindrical shells under pressure. Any solutions of the buckling problem can be
expanded as

ψ(X,θ)=
∑
an(θ)ηn(X) (4.10)

where an(θ) is an undetermined function which can be found by considering boundary condi-
tions.

5. Boundary conditions and the buckling bifurcation condition

All boundary conditions described at the two ends (X = ±L/2) can be derived from the
variational principle,Eq. (2.6). It iswell knownthat transverseboundaryconditions aregenerally
defined by the displacement or the internal force. In a symplectic system, they need to be
expressed in terms of Hamiltonian dual variables as:
— the clamped boundary condition

W = q1
∣∣∣
X=±L/2

=0 ∂XW = ∂Xq1
∣∣∣
X=±L/2

=0 (5.1)

— the simply supported boundary condition

W = q1
∣∣∣
X=±L/2

=0 ∂2XW = ∂
2
Xq1
∣∣∣
X=±L/2

=0 (5.2)

— the free boundary condition

QX =(1−ν)∂3Xq1+(2−ν)∂Xp2
∣∣∣
X=±L/2

=0 MX = νp2− (1−ν)∂2Xq1
∣∣∣
X=±L/2

=0

(5.3)

Meanwhile, the internal displacement and the force should also be satisfy the following four
in-plane conditions. For the displacement conditions U =0 and V =0, it needs to be replaced



Buckling of cylindrical shells under external pressure in a Hamiltonian system 647

by ∂2θU =0 and ∂θV =0, which can be deducted by considering Eq. (4.1). They are expressed
as:
—Condition 1

∂2θU =−(1+ν)∂3Xq3+
2+ν

β
∂Xp4+

1

H2
∂Xq1

∣∣∣
X=±L/2

=0

∂θV =(1+ν)∂
2
Xq3−

ν

β
p4+

1

H2
q1
∣∣∣
X=±L/2

=0
(5.4)

—Condition 2

∂2θU =(1+ν)∂
3
Xq3−

ν

β
∂Xp4+

1

H2
∂Xq1

∣∣∣
X=±L/2

=0

NXθ = ∂Xq4
∣∣∣
X=±L/2

=0
(5.5)

—Condition 3

NX =
p4
β
−∂2Xq3

∣∣∣
X=±L/2

=0 ∂θV = ∂
2
Xq3+

1

H2
q1
∣∣∣
X=±L/2

=0 (5.6)

—Condition 4

NX =
p4
β
−∂2Xq3

∣∣∣
X=±L/2

=0 NXθ = ∂Xq4
∣∣∣
X=±L/2

=0 (5.7)

By making eigenvectors Eqs. (4.5)-(4.8) satisfy the specific boundary conditions, a set of
eight homogeneous linear equations can be obtained as

Dc
1 =0 (5.8)

where c1 = {c11,c12, . . . ,c18} represents the unknown coefficients in the original variable q1. In
order that they have non-trivial solutions, the determinant of Eq. (5.8) should vanish. Then the
bifurcation condition can be given by

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

D
(i)
11(Ncr,n,−L/2) D

(i)
12(Ncr,n,−L/2) · · · D

(i)
18(Ncr,n,−L/2)

D
(i)
21(Ncr,n,−L/2) D

(i)
22(Ncr,n,−L/2) · · · D

(i)
28(Ncr,n,−L/2)

D
(j)
31 (Ncr,n,−L/2) D

(j)
32 (Ncr,n,−L/2) · · · D

(j)
38 (Ncr,n,−L/2)

D
(j)
41 (Ncr,n,−L/2) D

(j)
42 (Ncr,n,−L/2) · · · D

(j)
48 (Ncr,n,−L/2)

D
(i)
51(Ncr,n,L/2) D

(i)
52(Ncr,n,L/2) · · · D

(i)
58(Ncr,n,L/2)

D
(i)
61(Ncr,n,L/2) D

(i)
62(Ncr,n,L/2) · · · D

(i)
68(Ncr,n,L/2)

D
(j)
71 (Ncr,n,L/2) D

(j)
72 (Ncr,n,L/2) · · · D

(j)
78 (Ncr,n,L/2)

D
(j)
81 (Ncr,n,L/2) D

(j)
82 (Ncr,n,L/2) · · · D

(j)
88 (Ncr,n,L/2)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
8×8

=0 (5.9)

where i = 1,2,3; j = 1,2, . . . ,4 indicate three transverse boundaries and four in-plane boun-
daries, respectively. The critical load and the corresponding buckling mode can be determined
from Eq. (5.9) and Eqs. (4.5)-(4.8).

6. Buckling results and discussion

6.1. Results of cylindrical shells under symmetric boundary conditions

A dimensionless curvature parameter was introduced by Batdorf (1947) as Z =
=
√
1−ν2L2/H. In the following analysis, thickness H =h/R=1/405, Poisson’s ratio ν =0.3

is selected. Eight sets of symmetric boundary conditions are described as: C1: clamped edges
and Condition 1, C2: clamped edges and Condition 2, C3: clamped edges and Condition 3,



648 J. Sun et al.

C4: clamped edges and Condition 4, S1: simply supported edges and Condition 1, S2: simply
supported edges and Condition 2, S3: simply supported edges and Condition 3, S4: Simply
supported edges and Condition 4.

For some fixed geometrical parameters, the minimum critical pressure can always be deter-
mined for different eigenvalues µn = ni (n = ±1,±2, . . .). According to the uniform buckling
deflection, Eq. (4.1), it is obvious that the integer n indicates the number of circumferential
buckling waves. So the corresponding buckling modes can also be referred to as the n-th order
bucklingmodes (Xu et al., 2006). Variations of the minimum critical pressures Ncr determined
by bifurcation condition Eq. (5.9) with Z are displayed in Fig. 2. It is seen there that themini-
mumbuckling loads decrease rapidly with the increase of Z. For an extremely short cylindrical
shell with Z < 5, the minimum critical loads of cases C1-C4, calculated for Sort 4, is smaller
than that of cases S1-S4, and the in-plane boundary conditions have tiny influence on the mi-
nimum critical pressure. For Z greater than 5, the effect of the in-plane boundary conditions
becomes more significant than that of transverse boundary conditions, and the results of C1,
C2, S1, S2, belonging to Sort 3, are greater than those for C3, C3, S3, S3. The longer the
cylindrical shell is, the more distinctly this discrepancy becomes. Meanwhile, variations of the
corresponding circumferential waves with the geometrical parameter Z are presented in Fig. 3.
In order to distinguish between different cases, a dimensionless wave factor N =Ln/π is intro-
duced. The tendency of variations of the wave factors N with Z is similar to that of variations
of Ncr with Z.

Fig. 2.N
cr
vs. Z under symmetric boundary conditions: (a) C1,C3, S1 and S3; (b) C2,C4, S2

and S4

Fig. 3.N vs. Z under symmetric boundary conditions: (a) C1,C3, S1 and S3; (b) C2,C4, S2 and S4

Corresponding to Fig. 2 and Fig. 3, the buckling modes of cylindrical shells with Z = 103

under each boundary condition are illustrated in Fig. 4. It is shown in Fig. 4 that the in-



Buckling of cylindrical shells under external pressure in a Hamiltonian system 649

plane and transverse boundary conditions have no distinct influence on the symmetric buckling
deflections, and there is only one half-wave in the axial direction. In addition, the buckling
modes of cylindrical shells subjected to typical boundary conditions C1 are displayed in Fig. 5
when Z is equal to 20, 50, 100, 200, 500, respectively. It is found that regardless how long is
the cylindrical shell, the waveforms in the axial direction do not exhibit any change. But the
number of circumferential waves should decrease dramatically with the increasing Z.

Fig. 4. Buckling modes for Z =103 under symmetric boundary conditions

Fig. 5. Buckling modes vs. Z under boundary condition C1

Fig. 6. The first eight branches vs. Z for the order n=10

For fixed eigenvalues µn =ni (n=±1,±2, . . .), a series of critical pressures Ncr should be
determined by bifurcation condition Eq. (5.9) and can bemarked as different branches, such as
the first branch for m=1, etc. Figure 6 represents variation of the first eight branches versus Z
for the number of waves n=10 under boundary conditions C1. From these curves, it is shown
that each branch decreases to some distinct value rapidly with the increase of Z. The higher
the branch number is, the greater the critical pressure becomes.



650 J. Sun et al.

Figure 7 presents the variations of the first branches for the orders n=5, 10, 15, 20, 25, 30
versus Z under typical boundary condition C1.With the increase of Z, the first branches, be-
longing to different orders, would intersect each other. And intersections between these branches
indicate that cylindrical shells subjected to the same pressure should buckle into two different
modes.

Fig. 7. The first branches for the orders n=5, 10, 15, 20, 25, 30 vs. Z under C1

The relation between theminimumcritical pressures Nθ and the dimensionless thickness H
of cylindrical shells with different lengths L = 0.25, 0.5, 1,2 is presented in Fig. 8a. Here,
the dimensionless parameter Nθ = H

2Ncr is introduced to repersent the actual pressure. The
corresponding circumferential waves are also shown in Fig. 8b. With the increase of thickness,
the critical prssures become more higher, and the circumferential waves are lower. And this
tendency is more clear for a shorter shell.

Fig. 8. Buckling versus thickness for different lengths: (a) the minimum buckling loads,
(b) circumferential waves

6.2. Results of cylindrical shells under non-symmetric boundary conditions

Assume now that H = 1/200 and ν = 0.3. The results are discussed for the following
boundary conditions:

• C-S1: (i) for X = L/2, clamped edge and Condition 1; (ii) for X = −L/2, simply
supported edge and Condition 1.

• C-F1: for X = L/2, clamped edge and Condition 1; for X = −L/2, free edge and
Condition 4.



Buckling of cylindrical shells under external pressure in a Hamiltonian system 651

In order to compare with the results mentioned above, Fig. 9 represents the variations of
the minimum critical pressures and the corresponding circumferential wave factors versus Z
under typical symmetric and non-symmetric boundary conditions, respectively. It is seen that
the curves have the same tendency with the variation of Z. The results of case C-F1 are far
less than the others due to the relaxation of the edge X =−L/2. The corresponding buckling
modes of the cylindrical shell for Z =100 are represented inFig. 10 under four typical boundary
conditions. It is shown in Fig. 10 that the buckling deformation of case C-F1 is totally different
from the others. And the buckling deformations present a “bell mouth” shape on the free edge.

Fig. 9. Buckling results for different non-symmetry boundary conditions: (a) the minimum buckling
loads; (b) circumferential waves

Fig. 10. Buckling modes of cylindrical shell for Z =100 under non-symmetric boundary conditions

7. Conclusion

For buckling analysis of cylindrical shells under symmetric boundary conditions, the minimum
critical pressures and circumferential waves of shorter cylindrical shells are mainly affected by
transverse boundary conditions.With the increase of length, the effect caused by in-plane boun-
dary conditions becomes more significant than that caused by transverse boundary conditions.
Regardless how long are the cylindrical shells, the waveforms of buckling modes always appear
one half-wave in the axial direction and are not influenced by any symmetric boundary con-
ditions. With regard to the results of cylindrical shells subjected to non-symmetric boundary
conditions, there is no obvious discrepancy with those mentioned above except for case C-F1.



652 J. Sun et al.

Theminimum critical pressures of cylindrical shells under free boundary conditions are far less
than those for other cases. And the buckling deflections disclose a “bell mouth” shape on the
free edge. All analytical results of the cylindrical shells under external pressure belong to Sort 3
and 4. For the influence of thickness, it is found that shorter shells should bemore significantly
affected by it. Furthermore, some other interesting buckling results are also discussed in detail.
The Hamiltonian system and the solution methodology developed here is effective and can be
extended to other engineering fields.

Acknowledgements

The supports of National Natural Science Foundation of China (No. 11372070), the National Basic

Research Program of China (973 Program,Grant No. 2014CB046803), are gratefully acknowledged.

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Manuscript received January 3, 2014; accepted for print January 23, 2014