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Dynamic torsional buckling of cylindrical shells in Hamiltonian system
Thin-Walled Structures 64 (2013) 23–30
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Dynamic torsional buckling of cylindrical shells in Hamiltonian system Xinsheng Xu a, Jiabin Sun a, C.W. Lim b,n a b
State Key Laboratory for Structural Analysis of Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, PR China
a r t i c l e i n f o
abstract
Article history: Received 25 June 2012 Received in revised form 10 October 2012 Accepted 28 November 2012 Available online 5 January 2013
By considering the effect of stress waves in a Hamiltonian system, this paper treats dynamic buckling of an elastic cylindrical shell which is subjected to an impact torsional load. A symplectic analytical approach is employed to convert the fundamental equations to the Hamiltonian canonical equations in dual variables. In a symplectic space, the critical torsion and buckling mode are reduced to solving the symplectic eigenvalue and eigensolution, respectively. The primary influence factors, such as the impact time, boundary conditions and thickness, are discussed in detail through some numerical examples. It is found that boundary conditions have limited influence except free boundary condition in the context of the scope in this paper. The localization of dynamic buckling patterns can be observed at the free end of the shell. The new analytical and numerical results serve as guidelines for safer designs of shell structures. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Cylindrical shell Dynamic buckling Hamiltonian system Stress wave Torsional impact
1. Introduction Cylindrical shells have been widely used as one of the basic components in many types of engineering structures. To improve the structural reliability and safety, it is of great significance to clarify the dynamic stability of cylindrical shells under various impact loads. Although dynamic buckling of cylindrical shells under an axial impact has been studied extensively, by contrast, dynamic torsional buckling of cylindrical shells receives relatively little attention due to the inherent mathematical difficulty. In some early theoretical studies, only approximate solutions were obtained for some special cases, such as that by Leyko and Spryszynski [1] in which dynamic buckling of a cylindrical shell subjected to a time-dependent torsion was analyzed by using an approximate energy method. Subsequently, by using the energy criterion proposed by Ru and Wang [2], Wang et al. [3] investigated dynamic stability of a plastic cylindrical shell subjected to impact torsion. In this analysis, the rigid-plastic linear hardening mode was adopted and the critical impact velocity was discussed in detail. More recently, Sofiyev et al. [4–6] conducted similar research for some new high-performance materials. Galerkin’s method combined with the Ritz type variation method or Lagrange– Hamilton type principle was applied to estimate the effect of configurations of constituent materials. To explore a different approach with respect to those published approximate methods, a general perturbation method was developed by Wang et al. [7] to study the impact torsional buckling for an elastic cylindrical shell with an arbitrary imperfection. The result showed that imperfect geometry significantly influences the static torsional buckling load. A
n
Corresponding author. Tel.: þ852 2788 7285; fax: þ852 2788 7612. E-mail address: [email protected] (C.W. Lim).
0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.11.003
brief review on the dynamic behavior of simple structures was reported by Jones [8] in 1989. A few excellent monographs which discuss the various aspects of dynamic bucking can be referred to Simitses [9] and Lindberg and Florence [10]. There exist some experimental studies in this regard [11,12]. A survey shows that increasing post-buckling deformation always begins to take place at an initial linear stage, especially for axial impact buckling of cylindrical shells [13,14]. Hence, it offers a good opportunity to capitalize this fact and to apply the small deformation theory for understanding some dynamic buckling phenomena. Accordingly, the study of impact torsional buckling of cylindrical shells can be converted to a bifurcation problem by studying the propagation of stress waves [15,16]. The buckling deformation in the disturbed region can be obtained based on the bifurcation buckling theory. Unfortunately, only some roughly theoretical analyses are conducted by authors and approximate solving methods are employed. Research works with rigorous analytical solutions to the titled problem for different boundary conditions have been limited. For static torsional bucking, Yamaki and Kodama [17] presented some accurate solutions by directly integrating the fundamental equations and eight symmetric boundary conditions were treated in the study. However, the solution method cannot constitute a rigorous approach and it cannot provide a uniform analysis process. In short, most of the analytical methods available can be regarded as some variations of the Lagrangian system approach. The basic equations are expressed as some higherorder partial differential equations and they are usually solved by assuming some shape functions in one or two spatial dimensions. This approach is commonly known as the semi-inverse approach. In an attempt to overcome the shortcomings of the semiinverse method, Zhong [18] presented a revolutionary solution
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X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
methodology, the symplectic analytical approach, for solving some basic problems in solid mechanics. Through the Legendre transformation, the governing equations are transformed into lower-order Hamiltonian canonical equations in dual variables. Later, Xu et al. [19] established a new symplectic system to investigate dynamic torsional buckling of clamped cylindrical shells. Nevertheless, the system is merely effective for a shell based on the Timoshenko’s model and it is not applicable to treat local buckling problems. To substantiate the unknown research area, therefore, this paper develops a new symplectic system to analyze dynamic torsional buckling of cylindrical shells with various boundary conditions by considering the stress wave propagation based on Donnell’s shell theory. Using numerical examples, some interesting insights into this problem are discussed in detail.
An elastic cylindrical shell with radius R, thickness h, length l, Young’s modulus E, Poisson’s ratio u and material density r, subjected to an impact torque is illustrated in Fig. 1. Adopting a circular cylindrical coordinate system, the constitutive relations are ( ) e N E 0 , ð1Þ ¼ w M 0 D where N¼{Nx, Ny, Nxy}T and M¼{Mx, My, Mxy}T are the stress resultants and stress couples per unit length, respectively. The elastic coefficient matrixes are given by 2 3 2 3 1 u 0 1 u 0 3 Eh 6 Eh 6 7 7 0 4 u 1 0 5: E¼ 4u 1 5, D ¼ 1u2 12 1u2 0 0 ð1uÞ=2 0 0 1u The geometric equations relating the strain vector e ¼{ex, ey, exy}T and curvature vector v ¼{kx, ky, kxy}T with the displacements are given by @u , @x
kx ¼
ey ¼
@2 w , @x2
1 @v w , R @y
ky ¼
1 @2 w 2
2
R @y
exy ¼ ,
1 @u @v , þ R @y @x
kxy ¼
1 @2 w : R @x@y
ð2Þ
Nx ¼
1 @ F 2
R @y
2
2
,
Ny ¼
@ F , @x2
2
N xy ¼
1 @ F : R @x@y
ð4Þ
where D ¼ Eh3/[12(1 u2)] and t is the time. Then, the Lagrange density function can be obtained through the variational principle and integration by parts as 1 @2 F @u @2 F 1 @v 1 @2 F 1 @u @v þ 2 w þ L¼ 2 2 R @x@y R @y @x R @y @x @x R @y !2 !2 2 2 1 @ F 1 @ F 1 @2 w 1 @2 w N 0 @w @w þ 2 2 þ D þ 2 2 xy 2 2 2Eh @x 2 R @x @y @x R @y R @y 2 2 2 1 @u 1 @v 1 @w rh rh rh , ð5Þ 2 @t 2 @t 2 @t
where Nxy 0 ¼ T~ xy = 2pR2 is the torsion stress resultant of the prebuckling membrane state. Using the Hamiltonian principle dL~ ¼ 0, the governing equations for this Donnell’s shell theory can be obtained as 8 ~ 2 1 < ddLF ¼ Eh r4 F þ 1R @@xw2 ¼ 0, ð6Þ 0 : dL~ ¼ Dr4 w 1 @2 F2 þ 2Nxy @2 w þ rh @2 w ¼ 0, R @xy R @x dw @t 2
2 where r2 ¼ @2 =@x2 þ 1=R2 @2 =@y is the Laplacian operator.
3. Hamiltonian system and dual equations For simplicity, the following dimensionless terms are defined:
Introducing a stress function F, we have 2
L~ ¼ Pe þ Pw V t Z t 1 Z 2p Z l" 1 @2 F @u @2 F 1 @v þ ¼ dt R dy w R2 @y2 @x @x2 R @y t0 0 0 0 !2 1 @2 F 1 @u @v 1 @ @2 F 1 @2 F þ þ R @x@y R @y @x 2Eh @x2 R2 @y2 0 !2 11 2 2 2 @ F 1 @ F 1 @ F AA 2ð1þ uÞ@ 2 2 2 R @x@y @x R @y 0 0 !2 !2 11 2 2 2 1 @ @2 w 1 @2 w @ w 1 @ w 1 @ w AA þ D þ 2 2 2ð1uÞ@ 2 2 2 2 R @x@y @x2 @x R @y R @y 2 2 2 # Nxy 0 @w @w 1 @u 1 @v 1 @w dx, rh rh rh @t 2 @t 2 @t R @x @y 2
2. Basic equations
ex ¼
the elastic potential energy, potential energy due to external load and kinetic energy as [19]
ð3Þ
The Lagrange functional, which is dependent on the incremental displacements (u, v, w) and stress function F, consists of
x u v , U¼ , V¼ , R R R a ¼ 12 1u2 , b ¼ aH2 ,
X¼
W¼
w , R
T cr ¼
Nxy 0 R2 , D
F¼
F 3
Eh
,
T¼
L¼ ct , R
l , R
g¼
H¼
h , R
rhc2 R2 D
,
ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ E=½2rð1 þ uÞ is the wave velocity [19]. By defining _, _ ¼ @W=@y and qXW¼ qW/qX, further taking x ¼ W _ and c ¼ F W the dimensionless Lagrange density function can be expressed as
2 2 1
€ þ 1 @2 W þ W _ @X W 1 gð@T W Þ2 : € L ¼ aW@2X F b @2X F þ F T cr W 2 2 X 2
ð8Þ Introducing a vector q ¼{q 1, q 2, q 3, q 4} T ¼{W, x, F, c} T, the corresponding dual vector is given by
Fig. 1. Geometry for a cylindrical shell subject to an impact torque.
9 8
_ T cr @X W > > & W þ@2X W > > > > > >
> > > > > > 2 € > > W þ@X W = < dL
: ¼ p¼ 2 _ > dq_ > > > b & F þ @X F > > > > > >
> > > > > > € þ @2 F ; : b F X
ð9Þ
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
The dual variables denote the equivalent transverse shear stress, bending moment, in-plane shear stress and normal stress, respectively. Then the Hamiltonian density function is expressed in dual variables as Hðq,pÞ ¼ pT q_ ðq,pÞLðq,pÞ ¼ p1 q2 þ
1 2 1 p þ p2 @2X q1 p3 q4 p 2 þp4 @2X q3 þ aq1 @2X q3 2 2 2b 4
T cr q2 @X q1 þ
1 g @T q1 2 : 2
ð10Þ
Further, using Legendre transformation, the Hamiltonian canonical equations are given by 8 < q_ ¼ ddHp , ð11Þ : p_ ¼ ddHq : Defining a state vector u ¼{qT, pT}T, Eq. (11) can be simplified as
u_ ¼ Hu
ð12Þ
where H is the Hamiltonian 2 0 1 0 6 @2 0 0 X 6 6 0 0 6 0 6 6 0 0 @2X 6 H¼6 6 g@2T T cr @X a@2X 6 6T @ 0 0 6 cr X 6 4 a@2X 0 0 0
0
operator matrix [20] given by 0 0 0 0 0 3
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
@2X
0
0
1
0
0
0
0
0
0
0
0
0
1
25
By solving Eq. (15) in the complex domain, the eigenvector can be expressed in a general form as
gn ¼ c1 el1 X þ c2 el2 X þ c3 el3 X þ c4 el4 X þ c5 el5 X þ c6 el6 X þc7 el7 X þc8 el8 X ,
ð16Þ
where ci ði ¼ 1, 2, . . ., 8Þ is a vector of unknown constants, which can be determined from the boundary conditions. The eight characteristic roots are lk(k¼1, 2, ....,8), which can be determined from Eq. (15). It is noted that n¼0 is a special case not included in the expression (16). For this case, because l ¼0 corresponds to a quadruple root, the eigenvector can be written as
g0 ¼ c1 el1 X þ c2 el2 X þ c3 el3 X þ c4 el4 X þ c5 X 3 þ c6 X 2 þ c7 X þ c8 :
ð17Þ
By referring to some specified boundary conditions, it can be proven that this is only a trivial solution which implies that shell buckling with an axisymmetric mode subject to a torsional load does not take place. This special case is neglected in the following analysis. The buckling solution can be expanded as
uðX, y, oÞ ¼
1 X 8 X
½an ðoÞcðknÞ ðoÞelk ðnÞX enyi
n¼1k¼1
7 7 7 0 7 7 1=b 7 7 7: 0 7 7 0 7 7 7 @2X 5 0
þ
1 X 8 X
½bn ðoÞckðnÞ ðoÞelk ðnÞX enyi
n¼1k¼1
1 X n¼1
0
The state vector u with dual variables constitutes a complete solving symplectic space.
1 X
an gn ðaÞ enyi þ
bn gn ðbÞ enyi ,
ð18Þ
n¼1
where an and bn are the unknown coefficients which can be determined from the boundary conditions while gn ðaÞ and gn ðbÞ are two symplectic adjoint eigenvectors corresponding to the a-group and b-group, respectively.
5. Boundary conditions and bifurcation conditions 4. Symplectic eigenvalues, eigenvectors and eigensolutions In a symplectic space, the solutions to Eq. (12) can be expressed as [20]
uðX, y,T Þ ¼ gðX,T Þemy
ð13Þ
where g and m represent eigenvector and eigenvalue, respectively. Substituting Eq. (13) into Eq. (12) yields an eigenvalue equation as HgðX,T Þ ¼ mgðX,T Þ:
ð14Þ
The continuity condition of a shell of revolution requires the periodicity condition u(X, 0, T) ¼ u(X, 2p, T) in the circumferential direction be satisfied. It can then be proved that mn ¼ niðn ¼ 0, 71, 72, . . .Þ. It is easily verified that the eigenvalues mn a0 always appear in pairs and they can be grouped into two groups: (a): mn, Re mn o0 or Re mn ¼0\Im mn o0 and (b): mnn ¼ mn . By defining a symplectic inner product o g1 , g2 4 ¼ R T q1 p2 qT2 p1 dX, the symplectic adjoint orthonormality relations can be established [20]. In the formulation, Fourier transformation is applied to transform from the time domain to the frequency domain, or F gn ðX, T ÞÞ ¼ gn ðX, oÞ where o is the frequency. For bifurcation buckling problem, o ¼0 [16]. Substituting the eigenvalue mn ¼ni into Eq. (14), the characteristic polynomial of Eq. (14) can be derived as 8
6
5
4
3
2
l þ al þ bl þ cl þ dl þ el þ f l þ g ¼ 0, 2
4
2
ð15Þ 2
3
where a¼ 4n , b¼2nTcri, c¼6n þ a /b go , d¼ 4n Tcri, e¼ 4n6 þ2n2go2, f¼2n5Tcri and g¼n8 n4go2.
Consider a long enough cylindrical shell fixed at one end X¼L and subjected to a uniform impact torque at the other end X¼0. Initially, stress waves begin to propagate along the axial direction with no wave reflection. The wave front X¼Xe divides the cylindrical shell into two regions: a disturbed region 1 and another undisturbed region 2. The torsion stress resultant of the shell is given by ( T cr 0 r X r X e , NXy ¼ ð19Þ 0 X e rX rL: At the impact end X¼ 0, the following transverse boundary conditions are specified. It should be noted that the boundary conditions can also be derived from the variational principle using dL~ ¼ 0. In a symplectic space, they are expressed in Hamilton dual variables. The clamped boundary condition is W ¼ q1 9X ¼ 0 ¼ 0,
@X W ¼ @X q1 9X ¼ 0 ¼ 0,
ð20Þ
the simply supported boundary condition is W ¼ q1 9X ¼ 0 ¼ 0,
@2X W ¼ @2X q1 9X ¼ 0 ¼ 0,
ð21Þ
and the free boundary condition is Q X ¼ ð1uÞ@3X q1 þ ð2uÞ@X p2 þ T cr q2 9X ¼ 0 ¼ 0, MX ¼ up2 ð1uÞ@2X q1 9X ¼ 0 ¼ 0:
ð22Þ
In addition to the transverse boundary conditions, the in-plane boundary conditions should also be satisfied. However, the con¨ ditions U ¼0 and V ¼0 need to be replaced by U¼0 and V˙ ¼0 [21], which can be easily expressed by dual variables. Then, these conditions are given by
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X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
Case 1.
Table 1 Comparison of symplectic solutions with classical solutions ks ¼ TcrL2/p2. Z¼
pffiffiffiffiffiffiffiffiffiffiffi 2 1u2 L =H
Present Yamaki [21]
10
20
50
10.22 10.22
12.47 12.47
19.52 19.53
100
200
30.23 30.28
49.00 49.04
500 95.96 96.02
1000 161.4 161.8
ð2 þ uÞ 1 @X p4 þ 2 @X q1 9X ¼ 0 ¼ 0, U€ ¼ ð1 þ uÞ@3X q3 þ b H u 1 V_ ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 9X ¼ 0 ¼ 0: b H
ð23Þ
Case 2. ð2 þ uÞ 1 U€ ¼ ð1 þ uÞ@3X q3 þ @X p4 þ 2 @X q1 9X ¼ 0 ¼ 0, b H NX y ¼ @X p4 9X ¼ 0 ¼ 0:
ð24Þ
Case 3. NX ¼
p4
b
@2X q3 9X ¼ 0 ¼ 0,
u 1 V_ ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 9X ¼ 0 ¼ 0: b H
ð25Þ
Case 4. NX ¼
p4
b
@2X q3 9X ¼ 0 ¼ 0,
NX y ¼ @X q4 9X ¼ 0 ¼ 0:
ð26Þ
At the wave front X¼Xe, the continuity condition requires that Fig. 2. Buckling loads with respect to Te.
q1 91 ¼ q1 92 ,
@X q1 91 ¼ @X q1 92 ,ð1 þuÞ@3X q3 þ
¼ ð1 þ uÞ@3X q3 þ
ð2 þ uÞ
b
@X p4 þ
1 H2
ð2 þ uÞ
b
@X p4 þ
1 H2
@X q1 91
@X q1 92 ,
u 1 u 1 ð1 þ uÞ@2X q3 p4 þ 2 q1 91 ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 92 : b b H H
ð27Þ
The solutions as given by Eq. (16) should satisfy any combinations of the transverse and in-plane boundary conditions at the location X¼0 and continuity condition at the location X¼ Xe. Then, a homogeneous system consisting of eight linear equations can be obtained as
De ¼ 0,
ð28Þ
where e ¼ fe1 , e2 , , e8 gT are the undetermined coefficients. For Eq. (28) to have a non-trivial solution, the determinant of the coefficient matrix vanishes, or jDj ¼ 0:
Fig. 3. Circumferential wave factors with respect to Te.
ð29Þ
Subsequently, the buckling load Tcr can be solved from Eq. (29) and buckling modes can be obtained from Eqs. (16) and (18), respectively.
Fig. 4. Buckling modes for different boundary conditions (Te ¼2.0).
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
6. Results and discussion First, to verify the correctness of the developed symplectic method, the problem is reduced to study the static torsional buckling of cylindrical shells under the clamped boundary condition. By taking the shell thickness H¼ 1/405 and Poisson’s
27
ratio u¼0.3, the present results are compared with those reported by Yamaki [21]. As shown in Table 1, it is evident that the obtained results agree well with the existing results. Then, taking another thicker shell with H¼1/100, the following nine cases of boundary conditions referring to the transverse constraints in Eqs. (20)–(22) and the in-plane constraints in
Fig. 5. Buckling modes with respect to Te (C-1).
Fig. 6. Buckling modes with respect to Te (S-1).
Fig. 7. Buckling modes with respect to Te (F).
28
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
Eqs. (23)–(26) are investigated: (i) C-1: clamped boundary and Case 1; (ii) C-2: clamped boundary and Case 2; (iii) C-3: clamped boundary and Case 3; (iv) C-4: clamped boundary and Case 4; (v) S-1: simply supported boundary and Case 1; (vi) S-2: simply supported boundary and Case 2; (vii) S-3: simply supported boundary and Case 3; (viii) S-4: simply supported boundary and Case 4; and (ix) F: free boundary and Case 4. With the propagation of stress waves, Region 1 of the shell begins to buckle when the wave front propagates at a critical time Te ¼cte/R. The main purpose of numerical examples is to find out any possible critical loads and buckling modes for a fixed critical time. In the actual calculation, there always exists a number of critical loads for different eigenvalues mn ¼ niðn ¼ 0, 71, . . .Þ. Here an integer n refers to the buckling circumferential wave number and the corresponding buckling mode is referred to as the nth order mode [19]. The minimum critical load can be picked out easily and its effect with critical time is shown in Fig. 2. The corresponding circumferential wave numbers are illustrated in Fig. 3. To distinguish the different curves, a dimensionless wave factor N ¼Ten/p is introduced. In Fig. 2, it is obvious that the buckling load decreases to a certain value with critical time increasing except for Case F. The in-plane boundary conditions do not have significant influence on the buckling loads. For the shorter critical time Te o0.5, relaxing the transverse boundary constraints causes buckling load and circumferential wavenumber to reduce. The buckling modes at Te ¼2 corresponding to the curves in Figs. 2 and 3 are displayed in Fig. 4 for which a torsional impact acts at the upper-end of the shell. The variations of buckling modes with respect to critical times are presented in Figs. 5–7. For the cases C-1 and S-1, boundary conditions do have a similar effect on the buckling modes and the circumferential
wavenumber decreases with increasing critical time. It is also noticed that the buckling modes always appear as uniform twisting along the axial direction regardless the location of the wave front. Case F is a special case where the buckling load is very much lower than the other cases and it maintains a fixed value as shown in Fig. 2. In addition, the shell always buckles into a special waveform with n ¼1 in the circumferential direction as shown in Fig. 7. With the variation of critical time, large axial deformation always concentrates near the impact end and this phenomenon is known as localized bucking modes. This localized buckling deformation is mainly due to the absorption of the impact energy caused by the external torsion in the vicinity of the impact end. For a fixed circumferential wave number n, a number of critical loads can also be derived from the bifurcation condition in Eq. (29) and they can be marked as different branches, such as the first branch m ¼1 shown in Fig. 8, etc. [19]. In the following discussion, only Case C-1 is considered. The first eight branches for n¼6 with boundary condition C-1 is shown in Fig. 8. It can be observed that all branches decrease rapidly to an almost identical value with increasing critical time. The critical torsional load is larger for a branch with higher branch number. The axial buckling modes of these branches for Te ¼ 2 corresponding to Fig. 8 are presented in Fig. 9. The modes show that buckling waves for higher branch number are more densely distributed along the axial direction. In addition, it is noticed that the branch number indicates the number of axial waves. The time-dependent first branches corresponding to n ¼ 71, 72, y, 7 8 for C-1 boundary conditions are presented in Fig. 10. The curves corresponding to pairing adjoint eigenvalus mn ¼ni and mnn ¼ ni are exactly coincident and they decrease with the increasing critical time. The minimum critical loads, shown in
Fig. 8. The first eight branches (n¼ 6).
Fig. 10. the first eight orders (m¼ 1).
Fig. 9. Buckling modes (n¼ 6).
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
29
Fig. 11. Buckling modes (m¼ 1): (a) modes of a-group. (b) Modes of b-group.
Fig. 12. Buckling loads with respect to Te for different H.
Fig. 13. Circumferential wave factors with respect to Te for different H.
Fig. 2, are obtained as an envelope of these curves. Moreover, there are intersections for higher-order buckling and these intersections indicate the presence of multiple buckling modes for a unique critical load. The buckling modes for the a-group and b-group corresponding to Fig. 10 for Te ¼2 are displayed in Fig. 11(a) and (b). It is observed that buckling modes with respect to the axial waveforms for different orders differ significantly. In fact, the a-group and b-group modes are identical because either
group maps to the other through a rotation with respect to the shell axis. Hence, it is sufficient to only study either of the groups. Finally, the effect of thickness H on the buckling load is illustrated in Fig. 12 and the circumferential wave factor with respect to time for different H is shown in Fig. 13 where T cr ¼ H2 T cr . From the figures, it clearly shows that the critical load reduces and the circumferential waves factor increases for a thinner cylindrical shell.
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X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
7. Conclusion This paper presents a new symplectic approach for dynamic buckling of cylindrical shells based on a small deformation Donnell’s shell theory. By studying propagation of stress waves, shell buckling in the disturbed region can be mathematically transformed into a bifurcation problem. Comparing to the classical analytical methods, the symplectic method established is very efficient in theory and computation. In the symplectic system, the shell governing equations are converted into the Hamilton canonical equations which are expressed in lower-order partial differential equations. Solution for the critical loads and buckling modes are reduced to solving for symplectic eigenvalues and eigensolutions, respectively, within the symplectic space. Some typical numerical examples are presented. The result indicates that buckling characteristics are not significantly affected by the in-plane boundary conditions while the transverse boundary conditions only have limited influence within a short period except free boundary condition. With the increase of critical time, the buckling load and circumferential wave number decrease to some specific values except a shell with a free end where its buckling load is very much lower than the others and localized bucking modes are observed. New buckling characteristics corresponding to different branches, orders and varying thickness are presented and discussed in detail.
Acknowledgments The supports of National Natural Science Foundation of China (No. 11072054), the National Basic Research Program of China (973 Program, Grant no. 2009CB724302), and the Open Funding of the State Key Laboratory of Structural Analysis for Industrial Equipment Dalian University of Technology (Project No. GZ1102) are gratefully acknowledged. References [1] Leyko J, Spryszynski S. Energy method of analysis of dynamic stability of a cylindrical shell subjected to torsion. Archiwum Mechaniki Stosowanej 1974;26(1):13–24.
[2] Ru CQ, Wang R. On two problems of plastic buckling of cylindrical shell under impulsive loading. Acta Mechanica Solida Sinica 1988;9(1):62–6. [3] Wang DY, Zhang SY, Yang GT. The impact torsional buckling for the rigid plastic cylindrical shell. Applied Mathematics and Mechanics (English Edition) 1993;14(8):693–8. ¨ [4] Sofiyev AH, Zerin Z, Turkmen M. The buckling of laminated cylindrical thin shells under torsion varying as a linear function of time. Turkish Journal of Engineering and Environmental Sciences 2003;27(4):237–45. [5] Sofiyev AH. Torsional buckling of cross-ply laminated orthotropic composite cylindrical shells subject to dynamic loading. European Journal of Mechanics A/Solids 2003;22:943–51. [6] Sofiyev AH, Schnack E. The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading. Engineering Structures 2004;26:1321–31. [7] Wang DY, Zhang SY, Yang GT. Impact torsional buckling of elastic cylindrical shells with arbitrary form imperfection. Applied Mathematics and Mechanics (English Edition) 1993;14(6):499–505. [8] Jones N. Recent studies on the dynamic plastic behavior of structures. Applied Mechanics Reviews 1989;42(4):97–115. [9] Simitses GJ. Instabilities of dynamic loaded structures. Applied Mechanics Reviews: -ASME 1987;40(10):1403–8. [10] Lindberg HL, Florence AL. Dynamic pulse buckling. Dordrecht, Boston: Martinus Nijhoff; 1987. [11] Wang DY, Ma HW, Yang GT. Studies on the torsional buckling of elastic cylindrical shells. Applied Mathematics and Mechanics 1992;13(3):211–5. [12] Ma HW, Zhang SY, Yang GT. Impact torsional buckling of plastic circular cylindrical shells: experimental study. International Journal of Impact Engineering 1999;22:531–42. [13] Coppa AP. On the mechanism of buckling of a circular cylindrical shell under longitudinal impact, Gen. El. Rep. R60SD494. Present at 10th international congress of applied mechanics; 1960. [14] Lindberg HE, Herbert RE. Dynamic buckling of a thin cylindrical shell under axial impact. Journal of Applied Mechanics 1966;33(1):105–12. [15] Ma HW, Wang DY, Yang GT. An experimental study on the static and impact torsional buckling of cylindrical shells. Acta Mechanica Solida Sinica 1993;6(4):429–37. [16] Han Q, Ma HW, Zhang SY, Yang GT. The dynamic buckling problem caused by propagation of stress wave in elastic cylindrical shells under impact torque. Applied Mathematics and Mechanics 1996;17(1):1–8. [17] Yamaki N, Kodama S. Buckling of circular cylindrical shells under torsion/ report 2. Reports of the Institute of High Speed Mechanics, Tohoku University 18;1967:121–42. [18] Zhong WX. Duality system in applied mechanics and optimal control. Boston: Kluwer Academic Publishers; 2004. [19] Xu XS, Ma JQ, Lim CW, Zhang G. Dynamic torsional buckling of cylindrical shells. Computers & Structures 2010;88:322–30. [20] Yao WA, Zhong WX, Lim CW. Symplectic elasticity. Singapore: World Scientific; 2009. [21] Yamaki N. Elastic stability of circular cylindrical shells. North-Holland: Amsterdam, New York, Oxford; 1984.
Contents lists available at SciVerse ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Dynamic torsional buckling of cylindrical shells in Hamiltonian system Xinsheng Xu a, Jiabin Sun a, C.W. Lim b,n a b
State Key Laboratory for Structural Analysis of Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, PR China Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, PR China
a r t i c l e i n f o
abstract
Article history: Received 25 June 2012 Received in revised form 10 October 2012 Accepted 28 November 2012 Available online 5 January 2013
By considering the effect of stress waves in a Hamiltonian system, this paper treats dynamic buckling of an elastic cylindrical shell which is subjected to an impact torsional load. A symplectic analytical approach is employed to convert the fundamental equations to the Hamiltonian canonical equations in dual variables. In a symplectic space, the critical torsion and buckling mode are reduced to solving the symplectic eigenvalue and eigensolution, respectively. The primary influence factors, such as the impact time, boundary conditions and thickness, are discussed in detail through some numerical examples. It is found that boundary conditions have limited influence except free boundary condition in the context of the scope in this paper. The localization of dynamic buckling patterns can be observed at the free end of the shell. The new analytical and numerical results serve as guidelines for safer designs of shell structures. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Cylindrical shell Dynamic buckling Hamiltonian system Stress wave Torsional impact
1. Introduction Cylindrical shells have been widely used as one of the basic components in many types of engineering structures. To improve the structural reliability and safety, it is of great significance to clarify the dynamic stability of cylindrical shells under various impact loads. Although dynamic buckling of cylindrical shells under an axial impact has been studied extensively, by contrast, dynamic torsional buckling of cylindrical shells receives relatively little attention due to the inherent mathematical difficulty. In some early theoretical studies, only approximate solutions were obtained for some special cases, such as that by Leyko and Spryszynski [1] in which dynamic buckling of a cylindrical shell subjected to a time-dependent torsion was analyzed by using an approximate energy method. Subsequently, by using the energy criterion proposed by Ru and Wang [2], Wang et al. [3] investigated dynamic stability of a plastic cylindrical shell subjected to impact torsion. In this analysis, the rigid-plastic linear hardening mode was adopted and the critical impact velocity was discussed in detail. More recently, Sofiyev et al. [4–6] conducted similar research for some new high-performance materials. Galerkin’s method combined with the Ritz type variation method or Lagrange– Hamilton type principle was applied to estimate the effect of configurations of constituent materials. To explore a different approach with respect to those published approximate methods, a general perturbation method was developed by Wang et al. [7] to study the impact torsional buckling for an elastic cylindrical shell with an arbitrary imperfection. The result showed that imperfect geometry significantly influences the static torsional buckling load. A
n
Corresponding author. Tel.: þ852 2788 7285; fax: þ852 2788 7612. E-mail address: [email protected] (C.W. Lim).
0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.11.003
brief review on the dynamic behavior of simple structures was reported by Jones [8] in 1989. A few excellent monographs which discuss the various aspects of dynamic bucking can be referred to Simitses [9] and Lindberg and Florence [10]. There exist some experimental studies in this regard [11,12]. A survey shows that increasing post-buckling deformation always begins to take place at an initial linear stage, especially for axial impact buckling of cylindrical shells [13,14]. Hence, it offers a good opportunity to capitalize this fact and to apply the small deformation theory for understanding some dynamic buckling phenomena. Accordingly, the study of impact torsional buckling of cylindrical shells can be converted to a bifurcation problem by studying the propagation of stress waves [15,16]. The buckling deformation in the disturbed region can be obtained based on the bifurcation buckling theory. Unfortunately, only some roughly theoretical analyses are conducted by authors and approximate solving methods are employed. Research works with rigorous analytical solutions to the titled problem for different boundary conditions have been limited. For static torsional bucking, Yamaki and Kodama [17] presented some accurate solutions by directly integrating the fundamental equations and eight symmetric boundary conditions were treated in the study. However, the solution method cannot constitute a rigorous approach and it cannot provide a uniform analysis process. In short, most of the analytical methods available can be regarded as some variations of the Lagrangian system approach. The basic equations are expressed as some higherorder partial differential equations and they are usually solved by assuming some shape functions in one or two spatial dimensions. This approach is commonly known as the semi-inverse approach. In an attempt to overcome the shortcomings of the semiinverse method, Zhong [18] presented a revolutionary solution
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X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
methodology, the symplectic analytical approach, for solving some basic problems in solid mechanics. Through the Legendre transformation, the governing equations are transformed into lower-order Hamiltonian canonical equations in dual variables. Later, Xu et al. [19] established a new symplectic system to investigate dynamic torsional buckling of clamped cylindrical shells. Nevertheless, the system is merely effective for a shell based on the Timoshenko’s model and it is not applicable to treat local buckling problems. To substantiate the unknown research area, therefore, this paper develops a new symplectic system to analyze dynamic torsional buckling of cylindrical shells with various boundary conditions by considering the stress wave propagation based on Donnell’s shell theory. Using numerical examples, some interesting insights into this problem are discussed in detail.
An elastic cylindrical shell with radius R, thickness h, length l, Young’s modulus E, Poisson’s ratio u and material density r, subjected to an impact torque is illustrated in Fig. 1. Adopting a circular cylindrical coordinate system, the constitutive relations are ( ) e N E 0 , ð1Þ ¼ w M 0 D where N¼{Nx, Ny, Nxy}T and M¼{Mx, My, Mxy}T are the stress resultants and stress couples per unit length, respectively. The elastic coefficient matrixes are given by 2 3 2 3 1 u 0 1 u 0 3 Eh 6 Eh 6 7 7 0 4 u 1 0 5: E¼ 4u 1 5, D ¼ 1u2 12 1u2 0 0 ð1uÞ=2 0 0 1u The geometric equations relating the strain vector e ¼{ex, ey, exy}T and curvature vector v ¼{kx, ky, kxy}T with the displacements are given by @u , @x
kx ¼
ey ¼
@2 w , @x2
1 @v w , R @y
ky ¼
1 @2 w 2
2
R @y
exy ¼ ,
1 @u @v , þ R @y @x
kxy ¼
1 @2 w : R @x@y
ð2Þ
Nx ¼
1 @ F 2
R @y
2
2
,
Ny ¼
@ F , @x2
2
N xy ¼
1 @ F : R @x@y
ð4Þ
where D ¼ Eh3/[12(1 u2)] and t is the time. Then, the Lagrange density function can be obtained through the variational principle and integration by parts as 1 @2 F @u @2 F 1 @v 1 @2 F 1 @u @v þ 2 w þ L¼ 2 2 R @x@y R @y @x R @y @x @x R @y !2 !2 2 2 1 @ F 1 @ F 1 @2 w 1 @2 w N 0 @w @w þ 2 2 þ D þ 2 2 xy 2 2 2Eh @x 2 R @x @y @x R @y R @y 2 2 2 1 @u 1 @v 1 @w rh rh rh , ð5Þ 2 @t 2 @t 2 @t
where Nxy 0 ¼ T~ xy = 2pR2 is the torsion stress resultant of the prebuckling membrane state. Using the Hamiltonian principle dL~ ¼ 0, the governing equations for this Donnell’s shell theory can be obtained as 8 ~ 2 1 < ddLF ¼ Eh r4 F þ 1R @@xw2 ¼ 0, ð6Þ 0 : dL~ ¼ Dr4 w 1 @2 F2 þ 2Nxy @2 w þ rh @2 w ¼ 0, R @xy R @x dw @t 2
2 where r2 ¼ @2 =@x2 þ 1=R2 @2 =@y is the Laplacian operator.
3. Hamiltonian system and dual equations For simplicity, the following dimensionless terms are defined:
Introducing a stress function F, we have 2
L~ ¼ Pe þ Pw V t Z t 1 Z 2p Z l" 1 @2 F @u @2 F 1 @v þ ¼ dt R dy w R2 @y2 @x @x2 R @y t0 0 0 0 !2 1 @2 F 1 @u @v 1 @ @2 F 1 @2 F þ þ R @x@y R @y @x 2Eh @x2 R2 @y2 0 !2 11 2 2 2 @ F 1 @ F 1 @ F AA 2ð1þ uÞ@ 2 2 2 R @x@y @x R @y 0 0 !2 !2 11 2 2 2 1 @ @2 w 1 @2 w @ w 1 @ w 1 @ w AA þ D þ 2 2 2ð1uÞ@ 2 2 2 2 R @x@y @x2 @x R @y R @y 2 2 2 # Nxy 0 @w @w 1 @u 1 @v 1 @w dx, rh rh rh @t 2 @t 2 @t R @x @y 2
2. Basic equations
ex ¼
the elastic potential energy, potential energy due to external load and kinetic energy as [19]
ð3Þ
The Lagrange functional, which is dependent on the incremental displacements (u, v, w) and stress function F, consists of
x u v , U¼ , V¼ , R R R a ¼ 12 1u2 , b ¼ aH2 ,
X¼
W¼
w , R
T cr ¼
Nxy 0 R2 , D
F¼
F 3
Eh
,
T¼
L¼ ct , R
l , R
g¼
H¼
h , R
rhc2 R2 D
,
ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ E=½2rð1 þ uÞ is the wave velocity [19]. By defining _, _ ¼ @W=@y and qXW¼ qW/qX, further taking x ¼ W _ and c ¼ F W the dimensionless Lagrange density function can be expressed as
2 2 1
€ þ 1 @2 W þ W _ @X W 1 gð@T W Þ2 : € L ¼ aW@2X F b @2X F þ F T cr W 2 2 X 2
ð8Þ Introducing a vector q ¼{q 1, q 2, q 3, q 4} T ¼{W, x, F, c} T, the corresponding dual vector is given by
Fig. 1. Geometry for a cylindrical shell subject to an impact torque.
9 8
_ T cr @X W > > & W þ@2X W > > > > > >
> > > > > > 2 € > > W þ@X W = < dL
: ¼ p¼ 2 _ > dq_ > > > b & F þ @X F > > > > > >
> > > > > > € þ @2 F ; : b F X
ð9Þ
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
The dual variables denote the equivalent transverse shear stress, bending moment, in-plane shear stress and normal stress, respectively. Then the Hamiltonian density function is expressed in dual variables as Hðq,pÞ ¼ pT q_ ðq,pÞLðq,pÞ ¼ p1 q2 þ
1 2 1 p þ p2 @2X q1 p3 q4 p 2 þp4 @2X q3 þ aq1 @2X q3 2 2 2b 4
T cr q2 @X q1 þ
1 g @T q1 2 : 2
ð10Þ
Further, using Legendre transformation, the Hamiltonian canonical equations are given by 8 < q_ ¼ ddHp , ð11Þ : p_ ¼ ddHq : Defining a state vector u ¼{qT, pT}T, Eq. (11) can be simplified as
u_ ¼ Hu
ð12Þ
where H is the Hamiltonian 2 0 1 0 6 @2 0 0 X 6 6 0 0 6 0 6 6 0 0 @2X 6 H¼6 6 g@2T T cr @X a@2X 6 6T @ 0 0 6 cr X 6 4 a@2X 0 0 0
0
operator matrix [20] given by 0 0 0 0 0 3
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
@2X
0
0
1
0
0
0
0
0
0
0
0
0
1
25
By solving Eq. (15) in the complex domain, the eigenvector can be expressed in a general form as
gn ¼ c1 el1 X þ c2 el2 X þ c3 el3 X þ c4 el4 X þ c5 el5 X þ c6 el6 X þc7 el7 X þc8 el8 X ,
ð16Þ
where ci ði ¼ 1, 2, . . ., 8Þ is a vector of unknown constants, which can be determined from the boundary conditions. The eight characteristic roots are lk(k¼1, 2, ....,8), which can be determined from Eq. (15). It is noted that n¼0 is a special case not included in the expression (16). For this case, because l ¼0 corresponds to a quadruple root, the eigenvector can be written as
g0 ¼ c1 el1 X þ c2 el2 X þ c3 el3 X þ c4 el4 X þ c5 X 3 þ c6 X 2 þ c7 X þ c8 :
ð17Þ
By referring to some specified boundary conditions, it can be proven that this is only a trivial solution which implies that shell buckling with an axisymmetric mode subject to a torsional load does not take place. This special case is neglected in the following analysis. The buckling solution can be expanded as
uðX, y, oÞ ¼
1 X 8 X
½an ðoÞcðknÞ ðoÞelk ðnÞX enyi
n¼1k¼1
7 7 7 0 7 7 1=b 7 7 7: 0 7 7 0 7 7 7 @2X 5 0
þ
1 X 8 X
½bn ðoÞckðnÞ ðoÞelk ðnÞX enyi
n¼1k¼1
1 X n¼1
0
The state vector u with dual variables constitutes a complete solving symplectic space.
1 X
an gn ðaÞ enyi þ
bn gn ðbÞ enyi ,
ð18Þ
n¼1
where an and bn are the unknown coefficients which can be determined from the boundary conditions while gn ðaÞ and gn ðbÞ are two symplectic adjoint eigenvectors corresponding to the a-group and b-group, respectively.
5. Boundary conditions and bifurcation conditions 4. Symplectic eigenvalues, eigenvectors and eigensolutions In a symplectic space, the solutions to Eq. (12) can be expressed as [20]
uðX, y,T Þ ¼ gðX,T Þemy
ð13Þ
where g and m represent eigenvector and eigenvalue, respectively. Substituting Eq. (13) into Eq. (12) yields an eigenvalue equation as HgðX,T Þ ¼ mgðX,T Þ:
ð14Þ
The continuity condition of a shell of revolution requires the periodicity condition u(X, 0, T) ¼ u(X, 2p, T) in the circumferential direction be satisfied. It can then be proved that mn ¼ niðn ¼ 0, 71, 72, . . .Þ. It is easily verified that the eigenvalues mn a0 always appear in pairs and they can be grouped into two groups: (a): mn, Re mn o0 or Re mn ¼0\Im mn o0 and (b): mnn ¼ mn . By defining a symplectic inner product o g1 , g2 4 ¼ R T q1 p2 qT2 p1 dX, the symplectic adjoint orthonormality relations can be established [20]. In the formulation, Fourier transformation is applied to transform from the time domain to the frequency domain, or F gn ðX, T ÞÞ ¼ gn ðX, oÞ where o is the frequency. For bifurcation buckling problem, o ¼0 [16]. Substituting the eigenvalue mn ¼ni into Eq. (14), the characteristic polynomial of Eq. (14) can be derived as 8
6
5
4
3
2
l þ al þ bl þ cl þ dl þ el þ f l þ g ¼ 0, 2
4
2
ð15Þ 2
3
where a¼ 4n , b¼2nTcri, c¼6n þ a /b go , d¼ 4n Tcri, e¼ 4n6 þ2n2go2, f¼2n5Tcri and g¼n8 n4go2.
Consider a long enough cylindrical shell fixed at one end X¼L and subjected to a uniform impact torque at the other end X¼0. Initially, stress waves begin to propagate along the axial direction with no wave reflection. The wave front X¼Xe divides the cylindrical shell into two regions: a disturbed region 1 and another undisturbed region 2. The torsion stress resultant of the shell is given by ( T cr 0 r X r X e , NXy ¼ ð19Þ 0 X e rX rL: At the impact end X¼ 0, the following transverse boundary conditions are specified. It should be noted that the boundary conditions can also be derived from the variational principle using dL~ ¼ 0. In a symplectic space, they are expressed in Hamilton dual variables. The clamped boundary condition is W ¼ q1 9X ¼ 0 ¼ 0,
@X W ¼ @X q1 9X ¼ 0 ¼ 0,
ð20Þ
the simply supported boundary condition is W ¼ q1 9X ¼ 0 ¼ 0,
@2X W ¼ @2X q1 9X ¼ 0 ¼ 0,
ð21Þ
and the free boundary condition is Q X ¼ ð1uÞ@3X q1 þ ð2uÞ@X p2 þ T cr q2 9X ¼ 0 ¼ 0, MX ¼ up2 ð1uÞ@2X q1 9X ¼ 0 ¼ 0:
ð22Þ
In addition to the transverse boundary conditions, the in-plane boundary conditions should also be satisfied. However, the con¨ ditions U ¼0 and V ¼0 need to be replaced by U¼0 and V˙ ¼0 [21], which can be easily expressed by dual variables. Then, these conditions are given by
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X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
Case 1.
Table 1 Comparison of symplectic solutions with classical solutions ks ¼ TcrL2/p2. Z¼
pffiffiffiffiffiffiffiffiffiffiffi 2 1u2 L =H
Present Yamaki [21]
10
20
50
10.22 10.22
12.47 12.47
19.52 19.53
100
200
30.23 30.28
49.00 49.04
500 95.96 96.02
1000 161.4 161.8
ð2 þ uÞ 1 @X p4 þ 2 @X q1 9X ¼ 0 ¼ 0, U€ ¼ ð1 þ uÞ@3X q3 þ b H u 1 V_ ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 9X ¼ 0 ¼ 0: b H
ð23Þ
Case 2. ð2 þ uÞ 1 U€ ¼ ð1 þ uÞ@3X q3 þ @X p4 þ 2 @X q1 9X ¼ 0 ¼ 0, b H NX y ¼ @X p4 9X ¼ 0 ¼ 0:
ð24Þ
Case 3. NX ¼
p4
b
@2X q3 9X ¼ 0 ¼ 0,
u 1 V_ ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 9X ¼ 0 ¼ 0: b H
ð25Þ
Case 4. NX ¼
p4
b
@2X q3 9X ¼ 0 ¼ 0,
NX y ¼ @X q4 9X ¼ 0 ¼ 0:
ð26Þ
At the wave front X¼Xe, the continuity condition requires that Fig. 2. Buckling loads with respect to Te.
q1 91 ¼ q1 92 ,
@X q1 91 ¼ @X q1 92 ,ð1 þuÞ@3X q3 þ
¼ ð1 þ uÞ@3X q3 þ
ð2 þ uÞ
b
@X p4 þ
1 H2
ð2 þ uÞ
b
@X p4 þ
1 H2
@X q1 91
@X q1 92 ,
u 1 u 1 ð1 þ uÞ@2X q3 p4 þ 2 q1 91 ¼ ð1 þ uÞ@2X q3 p4 þ 2 q1 92 : b b H H
ð27Þ
The solutions as given by Eq. (16) should satisfy any combinations of the transverse and in-plane boundary conditions at the location X¼0 and continuity condition at the location X¼ Xe. Then, a homogeneous system consisting of eight linear equations can be obtained as
De ¼ 0,
ð28Þ
where e ¼ fe1 , e2 , , e8 gT are the undetermined coefficients. For Eq. (28) to have a non-trivial solution, the determinant of the coefficient matrix vanishes, or jDj ¼ 0:
Fig. 3. Circumferential wave factors with respect to Te.
ð29Þ
Subsequently, the buckling load Tcr can be solved from Eq. (29) and buckling modes can be obtained from Eqs. (16) and (18), respectively.
Fig. 4. Buckling modes for different boundary conditions (Te ¼2.0).
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
6. Results and discussion First, to verify the correctness of the developed symplectic method, the problem is reduced to study the static torsional buckling of cylindrical shells under the clamped boundary condition. By taking the shell thickness H¼ 1/405 and Poisson’s
27
ratio u¼0.3, the present results are compared with those reported by Yamaki [21]. As shown in Table 1, it is evident that the obtained results agree well with the existing results. Then, taking another thicker shell with H¼1/100, the following nine cases of boundary conditions referring to the transverse constraints in Eqs. (20)–(22) and the in-plane constraints in
Fig. 5. Buckling modes with respect to Te (C-1).
Fig. 6. Buckling modes with respect to Te (S-1).
Fig. 7. Buckling modes with respect to Te (F).
28
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
Eqs. (23)–(26) are investigated: (i) C-1: clamped boundary and Case 1; (ii) C-2: clamped boundary and Case 2; (iii) C-3: clamped boundary and Case 3; (iv) C-4: clamped boundary and Case 4; (v) S-1: simply supported boundary and Case 1; (vi) S-2: simply supported boundary and Case 2; (vii) S-3: simply supported boundary and Case 3; (viii) S-4: simply supported boundary and Case 4; and (ix) F: free boundary and Case 4. With the propagation of stress waves, Region 1 of the shell begins to buckle when the wave front propagates at a critical time Te ¼cte/R. The main purpose of numerical examples is to find out any possible critical loads and buckling modes for a fixed critical time. In the actual calculation, there always exists a number of critical loads for different eigenvalues mn ¼ niðn ¼ 0, 71, . . .Þ. Here an integer n refers to the buckling circumferential wave number and the corresponding buckling mode is referred to as the nth order mode [19]. The minimum critical load can be picked out easily and its effect with critical time is shown in Fig. 2. The corresponding circumferential wave numbers are illustrated in Fig. 3. To distinguish the different curves, a dimensionless wave factor N ¼Ten/p is introduced. In Fig. 2, it is obvious that the buckling load decreases to a certain value with critical time increasing except for Case F. The in-plane boundary conditions do not have significant influence on the buckling loads. For the shorter critical time Te o0.5, relaxing the transverse boundary constraints causes buckling load and circumferential wavenumber to reduce. The buckling modes at Te ¼2 corresponding to the curves in Figs. 2 and 3 are displayed in Fig. 4 for which a torsional impact acts at the upper-end of the shell. The variations of buckling modes with respect to critical times are presented in Figs. 5–7. For the cases C-1 and S-1, boundary conditions do have a similar effect on the buckling modes and the circumferential
wavenumber decreases with increasing critical time. It is also noticed that the buckling modes always appear as uniform twisting along the axial direction regardless the location of the wave front. Case F is a special case where the buckling load is very much lower than the other cases and it maintains a fixed value as shown in Fig. 2. In addition, the shell always buckles into a special waveform with n ¼1 in the circumferential direction as shown in Fig. 7. With the variation of critical time, large axial deformation always concentrates near the impact end and this phenomenon is known as localized bucking modes. This localized buckling deformation is mainly due to the absorption of the impact energy caused by the external torsion in the vicinity of the impact end. For a fixed circumferential wave number n, a number of critical loads can also be derived from the bifurcation condition in Eq. (29) and they can be marked as different branches, such as the first branch m ¼1 shown in Fig. 8, etc. [19]. In the following discussion, only Case C-1 is considered. The first eight branches for n¼6 with boundary condition C-1 is shown in Fig. 8. It can be observed that all branches decrease rapidly to an almost identical value with increasing critical time. The critical torsional load is larger for a branch with higher branch number. The axial buckling modes of these branches for Te ¼ 2 corresponding to Fig. 8 are presented in Fig. 9. The modes show that buckling waves for higher branch number are more densely distributed along the axial direction. In addition, it is noticed that the branch number indicates the number of axial waves. The time-dependent first branches corresponding to n ¼ 71, 72, y, 7 8 for C-1 boundary conditions are presented in Fig. 10. The curves corresponding to pairing adjoint eigenvalus mn ¼ni and mnn ¼ ni are exactly coincident and they decrease with the increasing critical time. The minimum critical loads, shown in
Fig. 8. The first eight branches (n¼ 6).
Fig. 10. the first eight orders (m¼ 1).
Fig. 9. Buckling modes (n¼ 6).
X. Xu et al. / Thin-Walled Structures 64 (2013) 23–30
29
Fig. 11. Buckling modes (m¼ 1): (a) modes of a-group. (b) Modes of b-group.
Fig. 12. Buckling loads with respect to Te for different H.
Fig. 13. Circumferential wave factors with respect to Te for different H.
Fig. 2, are obtained as an envelope of these curves. Moreover, there are intersections for higher-order buckling and these intersections indicate the presence of multiple buckling modes for a unique critical load. The buckling modes for the a-group and b-group corresponding to Fig. 10 for Te ¼2 are displayed in Fig. 11(a) and (b). It is observed that buckling modes with respect to the axial waveforms for different orders differ significantly. In fact, the a-group and b-group modes are identical because either
group maps to the other through a rotation with respect to the shell axis. Hence, it is sufficient to only study either of the groups. Finally, the effect of thickness H on the buckling load is illustrated in Fig. 12 and the circumferential wave factor with respect to time for different H is shown in Fig. 13 where T cr ¼ H2 T cr . From the figures, it clearly shows that the critical load reduces and the circumferential waves factor increases for a thinner cylindrical shell.
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7. Conclusion This paper presents a new symplectic approach for dynamic buckling of cylindrical shells based on a small deformation Donnell’s shell theory. By studying propagation of stress waves, shell buckling in the disturbed region can be mathematically transformed into a bifurcation problem. Comparing to the classical analytical methods, the symplectic method established is very efficient in theory and computation. In the symplectic system, the shell governing equations are converted into the Hamilton canonical equations which are expressed in lower-order partial differential equations. Solution for the critical loads and buckling modes are reduced to solving for symplectic eigenvalues and eigensolutions, respectively, within the symplectic space. Some typical numerical examples are presented. The result indicates that buckling characteristics are not significantly affected by the in-plane boundary conditions while the transverse boundary conditions only have limited influence within a short period except free boundary condition. With the increase of critical time, the buckling load and circumferential wave number decrease to some specific values except a shell with a free end where its buckling load is very much lower than the others and localized bucking modes are observed. New buckling characteristics corresponding to different branches, orders and varying thickness are presented and discussed in detail.
Acknowledgments The supports of National Natural Science Foundation of China (No. 11072054), the National Basic Research Program of China (973 Program, Grant no. 2009CB724302), and the Open Funding of the State Key Laboratory of Structural Analysis for Industrial Equipment Dalian University of Technology (Project No. GZ1102) are gratefully acknowledged. References [1] Leyko J, Spryszynski S. Energy method of analysis of dynamic stability of a cylindrical shell subjected to torsion. Archiwum Mechaniki Stosowanej 1974;26(1):13–24.
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