Analytical research on dynamic buckling of thin cylindrical shells with thickness variation under axial pressure

Analytical research on dynamic buckling of thin cylindrical shells with thickness variation under axial pressure

Thin-Walled Structures 101 (2016) 213–221 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate...

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Thin-Walled Structures 101 (2016) 213–221

Contents lists available at ScienceDirect

Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Full length article

Analytical research on dynamic buckling of thin cylindrical shells with thickness variation under axial pressure Haigui Fan, Zhiping Chen n, Jian Cheng, Song Huang, Wenzhuo Feng, Lige Liu Institute of Process Equipment, Department of Chemical and Biological Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 11 March 2015 Received in revised form 8 January 2016 Accepted 11 January 2016

This paper focuses on the dynamic buckling of thin cylindrical shells with arbitrary axisymmetric thickness variation under time dependent axial pressure. Based on the derivation of stability and compatibility equations of variable thickness cylindrical shells under dynamic external pressure by Aksogan and Sofiyev, the corresponding stability and compatibility equations of thin cylindrical shells with arbitrary axisymmetric thickness variation under dynamic axial pressure are obtained and expressed in nondimensional form. Combining the small parameter perturbation method, Fourier series expansion and the Sachenkov–Baktieva method, analytical formulas of the critical buckling load of thin cylindrical shells with arbitrary axisymmetric thickness variation under axial pressure that varies as a power function of time are obtained. Two cases of thickness variation are introduced to research the critical dynamic buckling load with the present formulas. Effects of thickness variation parameters and loading speed of dynamic axial pressure on the critical buckling load are discussed. The method is also applied to determine the critical dynamic buckling load of thin cylindrical shells with a classical cosine form thickness variation. Results revel that the thickness variation and pressure parameters play a major role in dictating the buckling capacity of thin cylindrical shells under dynamic axial pressure. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Critical buckling load Thickness variation Dynamic axial pressure Thin cylindrical shell Analytical formula

1. Introduction Cylindrical shells with thickness variation have gained more and more application in the oil, marine and aerospace fields in recent years [1–7]. Utilization of materials becomes more efficient and economic rationality is improved by using this structure. Dynamic buckling problems of cylindrical shells under axial pressure have been researched in previous literature. Tamura and Babcock [8] investigated the dynamic stability of an imperfect circular cylindrical shell subject to a step loading in the axial direction. The critical loads were determined by numerical integration of the equation of motion and compared with the static case. Huyan and Simitses [9] researched the dynamic stability of circular metallic and laminated cylindrical shells subjected to axial compression with geometrically imperfection. The dynamic critical loads were determined by the method of finite element and the equations of motion. Effects of axial pressure and structural imperfection on critical loads were discussed. An analytic solution for the stability behavior of cylindrical shells made of compositionally graded ceramic–metal materials under the axial n

Corresponding author. E-mail address: [email protected] (Z. Chen).

http://dx.doi.org/10.1016/j.tws.2016.01.009 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

compressive loads varying as a power function of time was provided by Sofiyev [10]. Effects of the variations of loading parameters and power of time in the axial load expression on the critical parameters of the shell were elucidated. Bisagni [11] dealt with dynamic buckling of fiber composite shells under impulsive axial compression using finite element method and experiment tests. It showed that the dynamic buckling loads strongly depended on the load duration and initial geometric imperfections. The dynamic buckling of thin isotropic thermoviscoplastic cylindrical shells compressed with a uniform axial impact was investigated analytically and numerically by Wei et al. [12] and the axisymmetric and non-axisymmetric dynamic buckling phenomena of cylindrical shells subjected to impacts of axial load was presented by Xu et al. [13]. Sofiyev et al. [14] studied the dynamic stability of orthotropic cylindrical shells with non-homogenous material properties under axial compressive load varying as a parabolic function of time. Expressions of the critical dynamic axial load and dynamic factor had been derived and effects of the axial loading parameter on the critical parameters had been researched. Semi-analytical solution for critical parameter values of dynamic axial compressed composite orthotropic cylindrical shell was obtained by Sofiyev et al. [15]. Huang and Han [16] researched the nonlinear dynamic buckling of functionally graded cylindrical shells subjected to axial load that varies linearly with regard to

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H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

time. Effects of the loading speed, dimension parameter and initial geometrical imperfection on critical condition were discussed. Besides the cylindrical shells under dynamic axial pressure, typical shell structures with variable thickness under dynamic external pressure have also attracted attention of some researchers. Aksogan and Sofiyev [17] and Sofiyev et al. [18] combined the Galerkin and Rize type variational method to obtain analytical solution of the critical static and dynamic loads, the corresponding wave numbers and dynamic factor of cylindrical shells with variable thickness under dynamic external pressure. Based on these methods, Sofiyev [19] and Sofiyev and Aksogan [20] researched buckling of conical shells with variable thickness under dynamic external pressure and found analytical expressions of the critical static and dynamic loads, the corresponding wave numbers, dynamic factor and critical stress impulse. Results showed that variation of thickness, external pressure and semivertex angle had appreciable effects on the critical dynamic factors. In addition, research on the buckling of cylindrical shells with non-uniform thickness subjected to static pressure had also been done by some researchers [21–29]. Review of the previous literature shows that researches are most about the cylindrical shells with constant thickness or made of composite materials under dynamic axial pressure, and cylindrical shells or conical shells with variable thickness under dynamic external pressure, which is perpendicular to the outside surface of the shells. Research on the cylindrical shells with thickness variation under dynamic axial pressure is quite limited. Thus we were prompted to present a theoretical method for obtaining analytical expressions of critical dynamic buckling load under axial pressure which takes on a power function of time, which can be applied to thin cylindrical shells with arbitrary axisymmetric thickness variation. Fig. 1. Geometry and coordinate of the axially compressed cylindrical shell.

2. Fundamental equations As shown in Fig. 1, a thin cylindrical shell of length L and radius R is subjected to dynamic axial compressive load varying as a power function of time, i.e.

P = P0t α

(1)

where P is the dynamic uniform pressure, P0 the loading speed, t the time and α the positive whole number power which express the time dependence of axial pressure. The linear strain–displacement relations of a thin cylindrical shell can be expressed as

∂u εx = ∂x κx = −

w ∂v εy = − R ∂y

∂ 2w ∂x2

κy = −

∂ 2w ∂y2

∂v ∂u γxy = + ∂y ∂x κxy = −

∂ 2w ∂x∂y

∂y

2

+

∂ 2εy ∂x

2



∂ 2γxy ∂x∂y

=−

1 ∂ 2w R ∂x2

(4)

∂ 2v ∂t 2

(5)

∂Nxy ∂x

∂Ny

+

∂y



2 ∂Nyz Ny ∂Nxz ∂ 2w ∂ 2w ∂ 2w 0 ∂ w + + + Nx0 2 + 2Nxy + N y0 2 = ρh 2 ∂x ∂y ∂x∂y R ∂x ∂y ∂t

(6)

where Nx , Ny , Nxy are membrane force components, Nxz , Nyz are

(2)

where u, v , w are displacements of the shell's middle surface along axial, circumferential and radial (positive inward) directions respectively. εx , εy , γxy are the strain components. κx , κy , κxy are curvature components of the middle plane. The following compatibility equation can be obtained by applying simple algebraic operation to Eq. (2)

∂ 2εx

∂Nxy ∂Nx ∂ 2u + =ρ 2 ∂x ∂y ∂t

(3)

Based on the Donnell simplified shell theory, the stability equations of a thin cylindrical shell under dynamic axial pressure are given as follows

0 are membrane force bending force components, Nx0 , N y0 , Nxy components in the initial stability state, and ρ is material density. For the uniform axial pressure as shown in Eq. (1), the membrane 0 force components satisfy N y0 = Nxy = 0, Nx0 = − hP0t α . The moment components are given as follows

(

Mx = D κx + υκ y

)

(

My = D κ y + υκx

3

)

Mxy = D ( 1 − υ)κxy

(7)

2

where D = Et /12(1 − υ ) is the flexural rigidity and υ is the Poisson ratio. The moment equilibrium equations are given by

∂Mxy ∂Mx + − Nxz = 0 ∂x ∂y ∂Mxy ∂x

+

∂My ∂y

− Nyz = 0

(8)

H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

Considering axisymmetry of the cylindrical shell, the thickness function h is supposed to just varies along the axial coordinate x . As a transformation from the regular Hooke's law, the following equation can be obtained

Nx − υNy

εx =

εy =

Eh(x)

Ny − υNx

γxy =

Eh(x)

(9)

The Airy stress function F(x,y), which is required to satisfy the following conditions, is introduced

∂ 2F ∂y2

Nx =

∂ 2F ∂x2

Ny =

Nxy = −

∂ 2F ∂x∂y

(10)

Because of the inequality relationship of the displacements u < < w, v < < w , the right side of Eqs. (4) and (5) satisfies ρ ∂ 2u/∂t 2 → 0, ρ ∂ 2v/∂t 2 → 0 [30]. Then Eqs. (4) and (5) can be satisfied automatically by substituting Eq. (10) into them. By substituting Eqs. (9) and (10) into Eq. (3), and then substituting Eqs. (1), (2), (7) and (8) into Eq. (6), one obtains the following linear governing equations of thin cylindrical shell with nonuniform thickness under dynamic axial pressure.

⎛ ∂ 4F ∂ 4F ∂ 4F ⎞ ∂h ⎛ ∂ 3F ∂ 3F ⎞ ⎟ − 2h ⎜ 3 − υ ⎟ h2⎜ 4 + 2 2 2 + 4 ∂ x ∂ x ∂ y ∂ x∂y2 ⎠ ∂y ⎠ ⎝ ∂x ⎝ ∂x ∂h ∂ 3F ∂x ∂x∂y2 ⎤ ⎡ ⎛ ⎞2 ∂h ∂ 2h ⎛ ∂ 2F ∂ 2F ⎞ Eh3 ∂ 2w +⎢ 2⎜ ⎟ − h 2 ⎥⎜ 2 − υ 2 ⎟ + =0 R ∂x2 ⎢⎣ ⎝ ∂x ⎠ ∂x ⎥⎦⎝ ∂x ∂y ⎠ − 2( 1 + υ)h

(11a)

⎛ ∂ 4w ∂ 4w ∂ 4w ⎞ 6Eh2 ⎜ 4 +2 2 2 + ⎟+ 4 ∂x ∂y ∂y ⎠ 12 1 − υ2 12 1 − υ ⎝ ∂x Eh3

(

2

⎛ ∂ 4ω L2 ∂ 4ω L4 ∂ 4ω ⎞ L2 ∂ 3ω ⎞ ∂H ⎛ ∂ 3ω ⎟ H3⎜ 4 + 2 2 2 2 + 4 4 ⎟ + 6H2 ⎜ 3 + 2 ∂ξ ⎝ ∂ξ R ∂ξ ∂θ R ∂ ξ ∂θ 2 ⎠ R ∂θ ⎠ ⎝ ∂ξ −

2( 1 + υ)Nxy Eh(x)

)

(

2 ⎤ ⎡ ⎛ L2 ∂ 2f L2 ∂ 2ω ⎞ ∂H ⎞ ∂ 2H ⎛ ∂ 2ω + ⎢ 6H ⎜ ⎟ + 3H2 2 ⎥⎜ 2 + v 2 2 ⎟ 2 ⎢⎣ ⎝ ∂ξ ⎠ Rh0 ∂ ξ R ∂θ ⎠ ∂ ξ ⎥⎦⎝ ∂ ξ

α α τ +P0tkr

h0L2 ∂ 2ω ρh0L4 ∂ 2ω H 2 + H 2 =0 2 D0 ∂ξ D0tkr ∂τ

f

ξ = 0,1

=0

ω ξ = 0,1 = 0

+ ρh

where q denotes the number of circumferential buckling waves. The numerical calculations show that, when the axial loading speed increases, the circumferential buckling waves decreases. In great values of axial loading speed, the circumferential buckling wave number q is equal to zero. Consequently, for great values of loading speed, an axisymmetric buckling occurs [10,14,15]. Considering the axisymmetric buckling ( q = 0 ), by substituting Eqs. (15a) and (15b) into Eqs. (13a) and (13b), the ordinary differential equations are obtained

(11b) −

f=

F D0

H=

h h0

ω=

w h0

ξ=

x L

θ=

y R

τ=

t tkr

∂ 4f¯ ∂ξ 4

− 2H

⎡ ⎛ 2 ⎤ 2¯ ⎞2 ∂H ∂ 3f¯ ⎢ 2⎜ ∂H ⎟ − H ∂ H ⎥ ∂ f + ∂ξ ∂ξ 3 ⎢⎣ ⎝ ∂ξ ⎠ ∂ξ 2 ⎥⎦ ∂ξ 2

(

)

Rh0

(

⎡ ⎛ 2 ⎤⎛ 2 2 2 ⎞ ⎞2 ∂H L2 ∂ 3f ⎢ 2⎜ ∂H ⎟ − H ∂ H ⎥⎜ ∂ f − v L ∂ f ⎟ + ∂ξ R2 ∂ ξ ∂θ 2 ⎢⎣ ⎝ ∂ξ ⎠ ∂ ξ 2 ⎥⎦⎝ ∂ ξ 2 R2 ∂θ 2 ⎠

)

12 1 − v2 L2 Rh0

H3

∂ 2ω =0 ∂ξ 2

∂ξ

4

+ 6H2

∂ 2ω¯ =0 ∂ξ 2

(16a)

2 ⎡ ⎛ 2 ⎤ 2 ∂H ∂ 3ω¯ ∂H ⎞ ∂ 2H ∂ ω¯ α α h0L τ H⎥ + ⎢ 6H ⎜ ⎟ + 3H2 2 + P0tkr 3 D0 ⎥⎦ ∂ξ 2 ∂ξ ∂ξ ∂ξ ⎣⎢ ⎝ ∂ξ ⎠

ρh0L4 ∂ 2ω¯ L2 ∂ 2f¯ H 2 =0 + 2 2 Rh0 ∂ξ D0tkr ∂τ

(16b)

3. Solution of the equations

⎛ ∂ 4f L2 ∂ 4f L4 ∂ 4f ⎞ L2 ∂ 3f ⎞ ∂H ⎛ ∂ 3f ⎟ H2⎜ +2 2 + 4 4 ⎟ − 2H ⎜ 3 − v 2 2 2 4 ∂ξ ⎝ ∂ξ R ∂ ξ ∂θ R ∂ ξ ∂θ 2 ⎠ R ∂θ ⎠ ⎝∂ξ

+

∂ 4ω¯

H3

(12)

where h0 is the nominal thickness of the shell, D0 is the flexural rigidity of the cylindrical shell with nominal thickness, and tkr is the critical buckling time. In view of the nondimensional parameters, the linear governing differential equations can be transformed by substituting Eq. (12) into Eqs. (11a) and (11b) to yield

−2( 1 + v)H

(14)

(15b)

H3

where E is the modulus of elasticity. To make solutions more general, the following nondimensional parameters are introduced

=0 ξ = 0,1

ω( ξ , θ , τ ) = ω¯ ( ξ , τ )sin qθ

12 1 − v2 L2

∂ 2w =0 ∂t 2

ξ = 0,1

∂ 2ω ∂ ξ2

(15a)

+

)

=0

f ( ξ , θ , τ ) = f¯ ( ξ , τ )sin qθ

E

(

∂ 2f ∂ ξ2

In view of the separation of variables, the solutions of stress function and radial displacement that satisfy the boundary condition are assumed as follows

∂ 3w ⎞ 1 ∂ 2F ∂h ⎛ ∂ 3w ⎜ 3 + ⎟− ∂x ⎝ ∂x ∂x∂y2 ⎠ R ∂x2 ⎡ ⎛ ⎞2 2 ⎤⎛ 2 2 ⎞ 2 ⎢ 6h⎜ ∂h ⎟ + 3h2 ∂ h ⎥⎜ ∂ w + υ ∂ w ⎟ + P0t αh ∂ w + 2⎥ 2 2 2 ⎢ ⎠ ⎝ x ∂ ⎝ ⎠ x x y x2 ∂ ∂ ∂ ∂ 12 1 − υ ⎣ ⎦

(13b)

The cylindrical shell is assumed to be simply supported, which satisfies [26]

H2

)

215

There are various forms of thickness variation along longitudinal direction in the cylindrical shells, such as linear thickness variation, parabolic thickness variation, classical trigonometric function thickness variation and local thickness variation. Because some variable thickness expressions are complicated, some parameters in the derivation process may not be analytically solved or expressed as analytical formulas if the variable thickness expressions are directly applied in the critical dynamic buckling load solution, leading to difficulties in the analytical expressions of critical dynamic buckling factors. Thus the small parameter perturbation method is used in this paper and the thickness function that varies in the axial direction arbitrarily is assumed to take on the following form ∞

(13a)

h = h0 + εh1 + ε2h2 + ... = h0 +

∑ εnhn n= 1

(17)

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H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

where ε(0 ≤ ε ≤ 1) is the nondimensional parameter indicating the magnitude of the thickness variation, hn(n ≥ 1) denotes unknown thickness variation function that determines the thickness of the shell. The variable thickness expression is decomposed into superposition of different terms and the critical dynamic factors corresponding to each term is solved respectively. The final critical dynamic buckling load can be obtained by superposition of the results corresponding to each term. The difficulties of applying complicated variable thickness expression directly in the solution process can be avoid by using this method and the analytical formulas of critical dynamic buckling load of the thin cylindrical shell with arbitrary axisymmetric thickness variation under axial pressure will be obtained. Eq. (17) can be transformed into a nondimensional form as

∂ 4f¯n ∂ξ

+ 4c 2r 2

4

∂ 4ω¯ n

∑ εnHn

(18)

n= 1

f¯ (ξ, τ ) and ω¯ (ξ, τ ) can also be expressed in term of ε as follows [26,31,32]

f¯ ( ξ , τ ) = f¯0 ( ξ , τ ) + εf¯1( ξ , τ ) + ε2f¯2 ( ξ , τ ) + …

∂ ξ4

2 2 D0T0 α ∂τ

ω¯ ( ξ , τ ) = ω¯ 0( ξ , τ ) + εω¯ 1( ξ , τ ) + ε2ω¯ 2( ξ , τ ) + …

∑ εnω¯ n( ξ, τ )

(19b)

n= 1

where f¯0 (ξ, τ ) and ω¯ 0(ξ, τ ) represent the solutions of the cylindrical shell with constant thickness h , f¯ (ξ, τ ) and ω¯ (ξ, τ ) represent the n

variation of stress function and radial displacement caused by the unknown thickness variation function hn . Owing to the arbitrariness of the parameter ε , f¯n (ξ, τ ) and ω¯ n(ξ, τ ) (n ≥ 0) must satisfy the following boundary conditions

f¯n ( 1, τ ) = 0 ∂ 2f¯n ∂ξ

ω¯ n( 0, τ ) = 0

I1 = 2H1

∂ ω¯ n ∂ ξ2

=0 ξ = 0,1

∑ε

∂ 4ω¯ 0 ∂ξ

4

+ 4c 2r 2

Tn

ξ = 0,1

(20)

(21)

2

∂ ω¯ 0 =0 ∂ξ 2

+ P0T0τ α

∂ξ

−2

4

3 2 ∂H1 ∂ f¯0 ∂ 2H1 ∂ f¯0 ∂ 2ω¯ − + 12c 2r 2H1 20 2 2 3 ∂ξ ∂ ξ ∂ξ ∂ξ ∂ ξ

(25a)

∂ ξ2 2 ρh0L4 ∂ 2ω¯ 0 ∂H1 ∂ 3ω¯ 0 2L2 T1 ∂ f¯0 − + H1 2 2 3 ∂ξ ∂ ξ Rh0 T0 ∂ ξ ∂τ D0T02

(25b)

⎞ ∂ 2ω¯ ⎛ T ⎞ ∂ 4ω¯ 0 ⎛ ∂ 2H1 h L2 h L2 0 + ⎜⎜ 3 2 + P0T0τ α 0 H1 + 0 P0T1τ α⎟⎟ Q 1 = ⎜ 3H1 + 1 ⎟ T0 ⎠ ∂ ξ 4 D D ⎝ ∂ ξ ⎝ ⎠ ∂ ξ2 0 0

+6

2 ∂H1 ∂ 3ω¯ 0 ρh0L4 ∂ 2ω¯ 0 L2 T1 ∂ f¯0 − + H1 2 ∂ξ ∂ ξ 3 Rh0 T0 ∂ ξ 2 D0T0 ∂τ

(25c)

when α > 2

Q 1 = 3H1

=0

where T0 represents the critical buckling time corresponding to nominal thickness h0 , Tn represents correction terms of critical buckling time corresponding to the unknown thickness variation function hn . Substituting Eqs. (18)–,(19b) and Eq. (21) into Eqs. (16a) and (16b), respectively, and collecting the like terms in ε0

∂ξ

∂ 4f¯0

∂ 4ω¯ 0 ∂ξ

4

ρh0L4 2 D0T0α

+6

H1

∂H1 ∂ 3ω¯ 0 ⎛ ∂ 2H1 h L2 ⎞ ∂ 2ω¯ 0 + ⎜⎜ 3 2 + P0T0τ α 0 H1⎟⎟ ∂ξ ∂ ξ 3 D0 ⎠ ∂ ξ 2 ⎝ ∂ξ

∂ 2ω¯ 0 ∂τ 2

(25d)

Firstly, the solutions of Eqs. (22a) and (22b) satisfying the boundary conditions are sought in the following form

f¯0 ( ξ , τ ) = A m (τ )sin mπξ

(26a)

ω¯ 0( ξ , τ ) = Bm(τ )sin mπξ

(26b)

n

n= 1

∂ 4f¯0 4

(24)

⎞ ⎛ h L2 2T ⎞ ∂ 4ω¯ 0 ⎛ ∂ 2H1 2h0L2 + ⎜⎜ 3 2 + P0T0τ α 0 H1 + Q 1 = ⎜ 3H1 + 1 ⎟ P0T1τ α⎟⎟ 4 T0 ⎠ ∂ ξ D0 D0 ⎝ ⎝ ∂ξ ⎠

+



T = T0 + εT1 + ε T2 + ... = T0 +

)

when α = 1

ω¯ n( 1, τ ) = 0

For convenience, we assume the critical buckling time satisfyα ing T = tkr . Then the solution can be expressed as follows 2

(

3 1 − υ2

The first term of In and Q n are provided here

2

2

c=

Rh0

when α = 2



f¯n ( 0, τ ) = 0

(23b)

L

r=

+6

n

+ Qn = 0

where

(19a)

n= 1

0

(23a)

∂ 2ω¯ 0

∑ εnf¯n ( ξ, τ )

= ω¯ 0( ξ , τ ) +

+ In = 0

2¯ 2 2 h0L2 ∂ 2ω¯ n ¯0 α h0L ∂ ω 2 ∂ fn τ − + r P T n 0 D0 ∂ ξ 2 D0 ∂ ξ 2 ∂ ξ2

ρh0L4 ∂ 2ω¯ n

+



= f¯0 ( ξ , τ ) +

∂ξ 2

+ P0T0τ α



H = 1 + εH1 + ε2H2 + ... = 1 +

∂ 2ω¯ n

∂ 2f¯ ρh0L4 ∂ 2ω¯ 0 h0L2 ∂ 2ω¯ 0 − r 2 02 + =0 2 2 2 D0 ∂ ξ ∂ξ D0T0 α ∂τ

then collecting the like terms in εn

where m is the number of half-waves along the shell length at buckling, Am (τ ) and Bm(τ ) are time-dependent amplitudes. Substituting Eqs. (26a) and (26b) into Eqs. (22a) and (22b) and eliminating the parameter Am (τ ), noting Eq. (24), we obtain

d2 Bm(τ ) dτ

(22a)

(22b)

2

⎡ D ( mπ )4 P T τ α( mπ )2 ⎤ α2 E ⎥T0 Bm(τ ) = 0 +⎢ 0 + − 00 2 2 4 ⎢⎣ ρh0L ⎥⎦ ρR ρL

(27)

The Sachenkov–Baktieva method is applied, i.e. multiplying the both sides of Eq. (27) by ∂Bm(τ ) /∂τ , then integrating with respect to τ first from 0 to τ and then form 0 to 1, we obtain

⎤ ⎡ Eh02( mπ )2 EL2 ⎥ ρL2 P0T0 = ⎢ J + 2 J1 + 2 2⎥ 0 ⎢ 12 1 − υ2 L2 R ( mπ ) ⎦ T0 α ( mπ )2 ⎣

(

)

(28)

H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

where J0 and J1 are defined as follows

J0 =

1

τ

2 ∫ ∫ ηαBm(η) 0 0

∫0 ⎡⎣ 1

J1 =



2

1

1

=

2

2∫ ∫ 0

(29)

Eq. (28) with respect to m , then the critical dynamic buckling load 0 of the cylindrical shell with constant thickness, Pkr , is obtained

h0L2

2

(31)

The solutions of Eqs. (23a) and (23b) satisfying the boundary conditions are sought in the Fourier series form ∞

(32a)

k=1

∑ Bk(τ )sin kπξ

(32b)

k=1

where k is an integer; Ak (τ ) and Bk(τ ) are time-dependent amplitudes. Substituting Eqs. (32a) and (32b) into Eqs. (23a) and (23b) yields ∞ 2 2

Ak (τ )sin kπξ − 4c r

k=1





4

k=1 ⎣

−P0Tnτ α

− P0T0τ α

(33a)

∞ ⎤ h0L2 2 2 kπ ) ⎥Bk(τ )sin kπξ + r 2 ∑ ( kπ ) Ak (τ )sin kπξ ( D0 ⎦ k=1

ρh0L4 h0L2 ( mπ )2Bm(τ )sin mπξ + 2 D0 D0T0 α



∑ k=1

d2 Bk(τ ) dτ 2

sin kπξ + Q n (33b)

=0

By multiplying the both sides of Eqs. (33a) and (33b) by sin mπξ respectively, then integrating ξ from 0 to 1, one obtains

∫0

1

In sin mπξ dξ = 0

(34a)

⎡ ⎤ h L2 ⎢ ( mπ )4 − P0T0τ α 0 ( mπ )2⎥Bm(τ ) + r 2( mπ )2A m (τ ) D ⎣ ⎦ 0 −P0Tnτ α

ρh 0 L

∫0

1

Q n sin mπξ dξ

1

In sin mπξ dξ (35)

2D0

P0Tn =

2

h0( mπ ) τ αBm(τ ) ⎡ 1 ×⎢ 2 ⎢L ⎣

∫0

1

Q n sin mπξ dξ −

1

2 ∫0

Rh0( mπ )

1

⎤ In sin mπξ dξ ⎥ ⎥ ⎦

(36)

The asymptotic formula of the critical dynamic buckling load of the thin cylindrical shell with arbitrary axisymmetric thickness variation under axial pressure which varies as a power function of time is thus given by ∞

Pkr = P0T0 +

∑ P0Tnϵn =

2D0J0 ( mπ )2

n= 1

+

h0L2

2D0 2

h0( mπ ) τ αBm(τ ) ⎡

1 2 n= 1 ⎣ L

∑ ⎢⎢

∫0

1

1

Q n sin mπξ dξ

2 ∫0

Rh0( mπ )

1

⎤ In sin mπξ dξ ⎥ϵn ⎥ ⎦

(37)

When the thickness variation expression of the shell is known, all of the parameters in Eq. (37) can be solved analytically and the critical dynamic buckling load can be obtained then.

4. Numerical results, discussions and comparisons

∑ ( kπ ) Bk(τ )sin kπξ + In = 0

( mπ )4 A m (τ ) − 4c 2r 2( mπ )2Bm(τ ) + 2

4

2

k=1

∑ ⎢ ( kπ )

∫0

2D0T0α

Considering Eq. (27), the left side of Eq. (35) becomes zero, and thus it can be transformed as follows





∑ ( kπ )

ρRL2h02( mπ )



∑ Ak (τ )sin kπξ

4

2

(30)

2 ⎤α ⎡ ⎡ D ( mπ )4 EL2 ⎤ 2J D0( mπ ) ⎥ = ρL2J1 J0 ⎢ 0 2 − 2 ⎥⎢ 0 ⎢⎣ h0L R ⎥⎦⎢⎣ P0h0L2 ⎥⎦



2D0T0α

2D0J0 ( mπ )2

By combining Eq. (28) and Eq. (30), the expression for determining dynamic buckling half-waves along the shell length can be obtained as follows

ω¯ n =

ρL +

2

f¯n =

2

T0α Bm(τ ) −

2



τ α dB (η) η Bm(η) dmη dηdτ 0

= P0T0 =

2

P0Tnτ α( mπ )

2

The minimization of P0T0 can be obtained by the derivative of

0 Pkr

2

2

dBm(η) dηdτ dη

dBm(τ ) ⎤ dτ ⎦

⎡ D ( mπ )4 P T τ α( mπ )2 ⎤ α2 E ⎥T0 Bm(τ ) +⎢ 0 + − 00 2 2 4 ⎢⎣ ρh0L ⎥⎦ ρR ρL

d2 Bm(τ )

∫0 ⎡⎣ Bm(τ )⎤⎦ dτ

217

ρh0L4 d2 Bm(τ ) h0L2 ( mπ )2Bm(τ ) + +2 2 2 D0 D0T0 α dτ

∫0

1

Q n sin mπξ dξ = 0 (34b)

By combining Eqs. (34a) and (34b), and substituting relevant parameters, we obtain

Consider the thickness variation form that has been discussed under external pressure by Aksogan and Sofiyev [17] and Sofiyev et al. [18]

⎛ x ⎞β h(x) = h0 − ( h0 − h1)⎜ ⎟ ⎝ L⎠

(38)

where h0 and h1 denote the thickness of each end of the cylindrical shell respectively, β is the power of thickness variation along the axial direction. As shown in Fig. 2, β = 1 and β = 2 correspond to the cases of linear and parabolic thickness variations respectively, which are more common in applications. Eq. (38) can be transformed into a nondimensional form as

⎛ h ⎞ H = 1 − ⎜ 1 − 1 ⎟ξ β h0 ⎠ ⎝

(39)

The parameters can be obtained as

ε=1− =0

h1 h0

H1 = − ξ β

Hn( n ≥ 2) (40)

The time-dependent amplitude Bm(τ ) satisfies the following initial boundary conditions [10,14,15,33]

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H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

⎤ ⎡ h L2( mπ )2 Q1 = ⎢ 0 P0T0τ − 3( mπ )4 ⎥Bm(τ )ξsin mπξ ⎥⎦ ⎢⎣ D0 +

⎤ h L2( mπ )2 2T1 ⎡ ⎢ ( mπ )4 + 4c 2r 4 − 0 P0T0τ ⎥Bm(τ )sin mπξ ⎥⎦ T0 ⎢⎣ D0

+6( mπ )3Bm(τ )cos mπξ −

ρh0L4 d2 Bm(τ ) T02D0

dτ 2

ξsin mπξ

(43b)

By substituting Eqs. (43a) and (43b) into Eq. (36) and noting the corresponding parameters, we obtain

⎫ ⎧ 4 4 2 ⎡ ⎤ ⎪ 3( mπ ) D0 4T1 ⎢ ( mπ ) D0 E E ⎥ ( mπ ) P0( 6T1 − T0)τ ⎪ ⎬ ⎨ + − + + 4 T0 ⎢⎣ ρh0L4 ⎪ ⎪ ρR 2 ρR2 ⎥⎦ ρL2 ⎭ ⎩ ρh 0 L T02Bm(τ ) +

d2 Bm(τ ) dτ 2

=0

(44)

By applying the Sachenkov–Baktieva method to Eq. (44), one can obtain the solution as follows

⎧ 3( mπ )4D0R 2 ⎫ J1 ρL2R2 + ⎨ + EL2⎬J0 T02 − ( mπ )2R2P0T03 ⎩ h 0L 2 ⎭ P0T1 = 4 2 ⎤ 6J0 T0 ⎡ ( mπ ) R D0 + EL2⎥ − 9( mπ )2R2T02 P0 ⎢ ⎣ h 0L 2 ⎦

(45)

The substitution of Eq. (45) into Eq. (37) furnishes the following critical dynamic buckling load

Pkr =

2D0J0 ( mπ )2 2

h0L

⎛ h ⎞ + ⎜1 − 1⎟ h0 ⎠ ⎝

⎧ 3( mπ )4D0R 2 ⎫ + EL2⎬J0 T02 − ( mπ )2R2P0T03 J1 ρL2R2 + ⎨ ⎩ h 0L 2 ⎭ × 4 2 ⎤ 4J0 T0 ⎡ ( mπ ) R D0 2 + EL ⎥ − 6( mπ )2R2T02 P0 ⎢ ⎣ h 0L 2 ⎦

(46)

Using the same method, the critical dynamic buckling load of the following three conditions will be derived Case B. (α = 1, β = 2): In this case, the axial pressure varies linearly as time and the thickness varies in parabolic form along the axial direction. The critical dynamic buckling load is derived as follows

Fig. 2. Schematic diagram of thickness variation.

Bm(τ )

τ= 0

=0

∂Bm(τ ) ∂τ

=0 τ= 0

(41)

The approximation function of Bm(τ ) is chosen in the following form

⎛ 53 ⎞ − τ⎟ Bm(τ ) = Ce50τ τ 2⎜ ⎝ 52 ⎠

(42)

where C is amplitude coefficient. Four conditions will be discussed here Case A. (α = 1, β = 1): In this case, the axial pressure varies linearly as time and the thickness varies linearly along the axial direction, substituting Eq. (40) into Eqs. (25a) and (25b), we obtain

I1 = 4c 2r 2( mπ )2Bm(τ )ξsin mπξ − 8c 2r 2( mπ )Bm(τ )ξ cos mπξ

(43a)

Pkr =

2D0J0 ( mπ )2 2

h0L

⎛ h ⎞ + ⎜1 − 1⎟ h0 ⎠ ⎝

⎧ 3( mπ )4D0R 2 ⎫ + EL2⎬J0 T02 − ( mπ )2R2P0T03 J1 ρL R + ⎨ ⎩ h 0L 2 ⎭ × 4 2 ⎤ 6J0 T0 ⎡ ( mπ ) R D0 2 + EL ⎥ − 9( mπ )2R2T02 P0 ⎢ ⎣ h 0L 2 ⎦ 2 2

(47)

Case C. (α = 2, β = 1): In this case, the axial pressure varies in parabolic form as time and the thickness varies linearly along the axial direction. The critical dynamic buckling load is found to be given by

H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

219

Table 1 Material properties and structural parameters. E (MPa)

ρ (kg/m3)

L (m)

R (m)

h0 (m)

υ

7.75  104

3100

0.792

0.36

0.002

0.3

Pkr =

2D0J0 ( mπ )2 h0L2

⎛ h ⎞ + ⎜1 − 1⎟ h0 ⎠ ⎝

⎧ 3( mπ )4D0R 2 ⎫ + EL2⎬T0J0 − ( mπ )2R2P0T02 J1 ρL2R2 + ⎨ ⎩ h 0L 2 ⎭ × 4 2 ⎡ ⎤ 2J0 ( mπ ) D0R + EL2⎥ − 4T0R2( mπ )2 P0 ⎢ ⎣ h 0L 2 ⎦

(48)

Case D. (α = 2, β = 2): In this case, the axial pressure and the thickness both vary in parabolic forms. The critical dynamic buckling load is given by Fig. 4. Critical dynamic buckling load vs thickness rate when α = 2.

Pkr =

2D0J0 ( mπ )2 h0L2

⎛ h ⎞ + ⎜1 − 1⎟ h0 ⎠ ⎝

⎧ 3( mπ )4D0R 2 ⎫ + EL2⎬T0J0 − ( mπ )2R2P0T02 J1 ρL2R2 + ⎨ ⎩ h 0L 2 ⎭ × 4 2 ⎤ 3J0 ⎡ ( mπ ) D0R 2 + EL ⎥ − 6T0R2( mπ )2 P0 ⎢ ⎣ h 0L 2 ⎦

(49)

The material properties and structural parameters of the cylindrical shell used in numerical computation is present in Table 1. When α = 1, three different values of loading speed, i.e. P0 = 7 × 104MPa/s, P0 = 9 × 104MPa/s, P0 = 11 × 104MPa/s are selected here. By substituting parameters of Table 1 and values of P0 into Eqs. (46) and (47), and in view of Eqs. (30) and (31), relation between the critical dynamic buckling load Pkr and thickness rate h1/h0 will be shown in Fig. 3. When α = 2, another three values of loading speed, i.e. are selected P0 = 0.112 × 106MPa/s2 , P0 = 0.448 × 106MPa/s2,

P0 = 1.68 × 106MPa/s2 here. By substituting parameters of Table 1 and values of P0 into Eqs. (48) and (49), and in view of Eqs. (30) and (31), relation between the critical dynamic buckling load Pkr and thickness rate h1/h0 will be obtained in Fig. 4. As it can be seen in Figs. 3 and 4, the critical dynamic buckling

load will decrease significantly along with the reduction of thickness rate. When the loading speed is constant, the critical dynamic buckling load of the cylindrical shell with a linear thickness variation is less than that with a parabolic thickness variation. As the thickness rate decreases, the reduction of the critical dynamic buckling load of the cylindrical shell with a linear thickness variation is faster. It indicates that the buckling capacity of a cylindrical shell under dynamic axial pressure is directly affected by the weakening of the thickness. When the thickness variation is unchanged, the critical dynamic buckling load, as well as the buckling capacity, will improve as the loading speed increases. However the critical dynamic buckling time tkr will reduce long with the increase of the loading speed. The difference of critical dynamic buckling loads between different loading speeds for a certain thickness variation form in Fig. 4 is smaller than that in Fig. 3, which means that the effects of loading speed on the critical dynamic buckling load become smaller when the power exponent of time increases. To verify the method presented herein, a comparison study has been made between the present results and those in previous research of Sofiyev [10] and Protsenko [34]. The cylindrical shell with constant thickness and parameters as in Table 1 is assumed to subject axial pressure that varies in parabolic form, i.e. α = 2. s s The dynamic factor Kd = Pkr /Pkr is introduced. Pkr = Eh/ 3(1 − υ2) R , is the static critical axial load. Results are shown in Table 2. A good match is achieved between the present results and those of previous researchers as shown in Table 2. Considering about the deficiency of analytic solutions and theoretical research on the buckling of variable thickness cylindrical shells made of isotropic materials under dynamic axial pressure, as an important part of the method presented in this paper, results in Table 2 verify the validity and accuracy of the present method preliminarily.

5. Application The critical dynamic buckling load of the cylindrical shells with Table 2 Comparison of the dynamic factor with previous results.

Fig. 3. Critical dynamic buckling load vs thickness rate when α = 1.

P0 (MPa) Kd (Ref. [34]) Kd (Ref. [10]) Kd (Present)

0.112  106

0.448  106

1.68  106

1.25

1.05

1.001

1.067

1.043

1.027

1.081

1.074

1.072

220

H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

classical trigonometric function thickness variation pattern under axial pressure that varies as a power function of time will be researched using the present method. The thickness variation considered is given by

the same way as in Case A

⎛ x⎞ h(x) = h0⎜ 1 − ε sin mπ ⎟ ⎝ L⎠

Pkr =

(50)

(51)

where

H1 = − sin mπξ

Hn( n ≥ 2) = 0

Case A. (α = 1): Substituting Eq. (52) into Eqs. (25a) and (25b), we obtain

I1 = − 8c 2r 2( mπ )2Bm(τ ) + 16c 2r 2( mπ )2Bm(τ )sin2mπξ

(53a)

⎡ h L2( mπ )2 ⎤ Q1 = ⎢ 0 P0T0τ − 6( mπ )4 ⎥Bm(τ )sin2mπξ ⎢⎣ ⎥⎦ D0 +

⎤ h L2( mπ )2 2T1 ⎡ ⎢ ( mπ )4 + 4c 2r 4 − 0 P0T0τ ⎥ ⎥⎦ T0 ⎢⎣ D0

Bm(τ )sin mπξ +6( mπ )4 Bm(τ )cos2mπξ −

ρh0L4 d2 Bm(τ ) T02D0

dτ 2

sin2

mπξ

(53b)

Substituting Eqs. (53a) and (53b) into Eq. (36) yields

⎧ 4 4 ⎡ ⎤ ⎪ 3( mπ ) D0 3( mπ )T1 ⎢ ( mπ ) D0 E E ⎥ ⎨ + − + 2 2 4 4 4T0 ⎢⎣ ρh0L ⎪ ρR ρR ⎥⎦ ⎩ ρh 0 L ⎡ 9( mπ ) ⎤ ( mπ )2P τ ⎫ 0 ⎪ 2 ⎬T B (τ ) T1 − T0⎥ +⎢ ⎢⎣ 8 ⎥⎦ ρL2 ⎪ 0 m ⎭ +

d2 Bm(τ ) dτ 2

=0

(54)

By applying the method of Sachenkov–Baktieva to Eq. (54), one obtains the critical dynamic buckling load as follows

(57)

The critical dynamic buckling load reduction factor is

(52)

Two different cases (i.e. α = 1 and α = 2) will be discussed here

h0L2 ⎧ 3( mπ )4D0R 2 ⎫ J1 ρL2R2 + ⎨ + EL2⎬J0 T0 − ( mπ )2R2P0T02 ⎩ h 0L 2 ⎭ + ε ⎫ ⎤ 3( mπ ) ⎧ J0 ⎡ ( mπ )4R 2D0 2 2 2 ⎬ ⎨ ⎢ EL 2 m T R + − ( π ) ⎥⎦ 0 8 ⎩ P0 ⎣ h 0L 2 ⎭

Eq. (50) can be transformed into a nondimensional form, i.e.

H = 1 − ε sin mπξ

2D0J0 ( mπ )2

λ=

Pkr 0 Pkr

=1+

J1 ρh 0L4R2 +

{ 3( mπ ) D R 4

0

2

}

+ Eh 0L4 J0 T0 − ( mπ )2h 0R2L2P0T02

3( mπ )3D0J0 ⎧ J0 ⎡ ( mπ )4R 2D0 ⎨ ⎢ 4 ⎩ P0 ⎣ h 0L2

⎫ ⎤ + EL2⎥ − 2( mπ )2T0R2⎬ ⎦ ⎭

ε

(58)

Three different values of loading speed, i.e. are P0 = 7 × 104MPa/s, P0 = 9 × 104MPa/s, P0 = 11 × 104MPa/s selected in this condition. Noting the parameters in Table 1, relations of the critical dynamic buckling load reduction factor λ and the thickness variation magnitude parameter ε are shown in Fig. 5. Fig. 5 shows that for the cylindrical shells with classical trigonometric function thickness variation pattern, the critical dynamic buckling load reduction factor will decrease as the thickness variation magnitude parameter increases, which verifies that the buckling capacity of the cylindrical shells under axial pressure is affected by the weakening of thickness. When the loading speed is constant, increase of the power exponent of time will lead to reduction of the critical dynamic buckling load reduction factor, meaning the lower axially pressurized buckling capacity of the cylindrical shell. When the power exponent remains unchanged, the critical dynamic buckling load reduction factor will become larger as the loading speed increases, while the critical dynamic buckling time will decrease at the same time. Analyses show that the method presented in this paper can be not only used to solve the linear and parabolic thickness variation problems, but also the relatively complicated thickness variation problems, such as the classical trigonometric function thickness variation. The critical dynamic buckling load can be obtained by this method with all the parameters solved in the analytical expressions form.

2D0J0 ( mπ )2

Pkr =

h0L2 ⎧ 3( mπ )4D0R 2 ⎫ + EL2⎬J0 T02 − ( mπ )2R2P0T03 J1 ρL2R2 + ⎨ ⎩ h 0L 2 ⎭ + ε ⎫ ⎤ 3 3( mπ ) ⎧ J0 T0 ⎡ ( mπ )4R 2D0 2 2 2 2 ⎨ ⎬ + − ( π ) EL m T R ⎥⎦ 2 0 4 ⎩ P0 ⎢ ⎣ h 0L 2 ⎭

(55)

The critical dynamic buckling load reduction factor λ due to thickness variation of the cylindrical shells is introduced as follows λ=

Pkr 0 Pkr

ε

=1+

}

J1 ρh 0L4R 2 + {3( mπ )4 D0R 2 + Eh 0L4 J0 T02 − ( mπ )2h 0R 2L2P0T03 ⎪ J 3( mπ )3D 0J0 ⎧ T ⎡ ( mπ )4R 2D 0 ⎨ 0 0⎢ ⎪ 2 2 ⎩ P 0 ⎣ h 0L

⎤ + EL2⎥ − ⎦

3 ( mπ )2T02R 2} 2

(56)

Case B. (α = 2): The critical dynamic buckling load is derived in

Fig. 5. Buckling load reduction factor vs thickness variation magnitude parameter.

H. Fan et al. / Thin-Walled Structures 101 (2016) 213–221

6. Conclusions By using the method of separation of variables, Fourier series expansion, the small parameter perturbation method and the method of Sachenkov–Baktieva, the analytical formulas of critical buckling loads of cylindrical shells with arbitrary axisymmetric thickness variation under dynamic axial pressure that varies as a power function of time are derived. The critical dynamic buckling loads of the cylindrical shells with three different thickness variations (i.e. the classical trigonometric function thickness variation, the linear and parabolic thickness variations) under different dynamic axial pressure are obtained by using these formulas. Results show that the critical dynamic buckling load, as well as the buckling capacity, of the cylindrical shells under dynamic axial pressure will decrease due to weakening of the thickness. The amplitude of the critical dynamic buckling load reduction becomes larger as the thickness variation increases. Coefficients of the dynamic axial pressure have significant influence on the critical dynamic buckling load. As the loading speed increases, the critical dynamic buckling load will improve while the critical dynamic buckling time will decrease. Increase of the power exponent will lead to the reduction of the critical dynamic buckling load.

Acknowledgments This work was supported by the Provincial Quality Supervision Research Program of Zhejiang (No. 20140239).

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