Buckling of the composite orthotropic clamped–clamped cylindrical shell loaded by transverse inertia forces

Composite Structures 95 (2013) 471–478

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Buckling of the composite orthotropic clamped–clamped cylindrical shell loaded by transverse inertia forces A.V. Lopatin a, E.V. Morozov b,⇑ a b

Department of Aerospace Engineering, Siberian State Aerospace University, Krasnoyarsk, Russia School of Engineering and Information Technology, University of New South Wales, Canberra, Australia

a r t i c l e

i n f o

Article history: Available online 31 July 2012 Keywords: Composite cylindrical shells Buckling Transverse inertia loading Galerkin method Critical acceleration

a b s t r a c t The paper is concerned with the buckling analysis of the composite orthotropic cylindrical shell with clamped edges subjected to inertia loading. The problem is characterised by a non-uniform pre-buckling stress state. To address this, the relevant governing system of differential equations with variable coefficients has been derived and solved using Galerkin method. As a result, the problem of determining the critical acceleration causing the buckling of shell is reduced to the solution of the corresponding generalised eigenvalue problem. Using the proposed approach, the critical accelerations are calculated for the glass-fibre reinforced shells with various dimensional and stiffness parameters. Results of calculations are compared with those based on finite-element modelling and analysis. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Overwhelming majority of the studies considering buckling of cylindrical shells has been performed for the cases when the shells are subjected to the surface loading, i.e. when the load is applied to the ends of the shell or to its cylindrical surface. The shell could be loaded with axial compressive load, transverse forces, bending or twisting moments, external pressure or various combinations of those. Numerous solutions of the related buckling problems can be found in the monographs published by Flügge [1], Timoshenko and Gere [2], Brush and Almroth [3], Yamaki [4], Volmir [5], Alfutov [6], Jones [7] and many others. However, there are practical engineering applications where the cylindrical shells are subjected to the body forces per unit volume, such as gravity, inertia loadings or electromagnetic forces. Buckling analysis of such shells could be of interest when designing this type of structural components. One of the examples is the analysis of the structure affected by the earthquake when the seismic load makes the shell move with the acceleration that could cause the loss of stability. Thin-walled electrically conductive shell can buckle while moving in the magnetic field. Inertia forces could cause buckling of the shell structures of the flying vehicle manoeuvring with large accelerations or decelerations. Note that to date, the number of the studies considering buckling of the thin-walled structures resulting from the application of the body forces is rather limited. Examples of such analyses are given by Lim et al. [8] and Lim and Ma [9]. The latter publication is ⇑ Corresponding author. E-mail address: [email protected] (E.V. Morozov). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.07.026

concerned with the elastic buckling of cylindrical shells under combined end pressure and axial body force. Such loadings are encountered when cylindrical shells are operating in a high-g environment such as launches of rockets and missiles driven by high-propulsive power. The related buckling problems have been solved for the short and long shells with free, simply supported and clamped ends using Goldenveizer–Novozhilov thin shell theory and the Ritz method. Analysis of publications shows that the solutions of buckling problems formulated for the cylindrical shells subjected to transverse body forces per unit volume are severely underrepresented (rather scarcely discussed in the literature) in the field under consideration. This paper is aimed to partially fill this gap considering the buckling analysis of composite cylindrical shell with fully clamped ends and loaded with the transverse inertia forces. The solution is based on the classical theory of the orthotropic laminated shells. The pre-buckling membrane stress resultants are found using the equations of the membrane theory of shells. The governing system of differential equations with variable coefficients representing the buckling problem is solved using Galerkin method. The axial displacements are approximated with the first vibration mode clamped-camped beam functions. The ordinary trigonometric series are employed to represent the displacements in hoop direction. The buckling problem is reduced to the calculation of the minimum eigenvalue for the homogeneous system of linear algebraic equations obtained as a result of application of Galerkin procedure. The critical accelerations have been found solving the numerical examples for the composite and isotropic shells. The obtained results have been verified by comparisons with the solutions obtained from the finite-element analyses.

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2. Problem formulation Consider orthotropic composite cylindrical shell of length l. The middle surface of the shell with the radius R is referred to the curvilinear coordinate frame a, b, and c as shown in Fig. 1. As can be seen, the coordinate a is parallel to the axis of the cylinder, b is the hoop coordinate, and c is normal to the middle surface. The ends of the shell a = 0 and a = / are fully clamped. The shell is subjected to the transverse inertia load orthogonal to the shell axis. The load can be treated as the uniformly distributed gravity force p shown in Fig. 2. This load has a dimension of pressure and is calculated as follows

p ¼ Bq nc g

ð1Þ

where Bq is the mass of the unit area of the shell material, nc is the transverse g-force, and g = 9.8 m/s2. Under this loading, the top part of the shell is compressed and can buckle when the g-force (or acceleration ncg) reaches the critical value. Buckling of the shell under consideration can be described by the following classical theory of orthotropic shells [10], which includes: linearised buckling equations

@Na @Nab ¼0 þ @a @b @Nab @Nb 1 @M ab 1 @M b þ þ þ ¼0 R @b @a @b R @ a

Fig. 2. Uniformly distributed gravity force p.

constitutive equations

in-plane strains of the middle surface; ka, kb, kab bending deformations of the middle surface; xa, xb rotation angles of the elements tangent to the coordinate lines a and b; u, v and w in-plane displacements and deflection of the middle surface; B11, B12, B22, B33(B21 = B12) and D11, D12, D22, D33(D21 = D12) membrane and bending stiffnesses of the shell wall; N0a ; N0b ; N 0ab membrane pre-buckling forces caused by g-force nc. The boundary conditions at the fully clamped edges a = 0 and a = l are satisfied if

N a ¼ B11 ea þ B12 eb

M a ¼ D11 ka þ D12 kb

u¼0

N b ¼ B21 ea þ B22 eb

Mb ¼ D21 ka þ D22 kb

@ 2 Ma @ 2 M ab @ 2 Mb Nb þ2  þ  @ a2 @ a@b R @b2   @ xa @ xa @ xb @ xb  N0b  N0ab N 0a þ ¼0 @a @b @a @b

N ab ¼ B33 eab

ð2Þ

ð3Þ

M ab ¼ D33 kab

strain–displacement relationships

@u @2w ka ¼  2 @a @a @v w @2w 1 @v kb ¼  2 þ ¼ þ R @b @b R @b

ea ¼ eb

@u @ v @2w 1 @v eab ¼ þ kab ¼ 2 þ @b @ a @ a@b R @ a @w v @w xa ¼  xb ¼  @a R @b

v ¼0

w¼0

@w ¼0 @a

ð5Þ

Clearly, for the loading case under consideration, the pre-buckling stress state of the shell is not uniform. Hence, the membrane forces N0a ; N0b , and N0ab are some functions of coordinates a and b. In order to find these forces, the membrane theory of shells is employed [10]. According to this theory, the membrane forces are related through the following equations of equilibrium: 0

ð4Þ

in which Na, Nb, Nab are the membrane stress resultants; Ma, Mb, Mab bending and twisting moments; ea, eb, eab normal and shear

@N0a @Nab þ þ pa ¼ 0 @a @b

ð6Þ

@N0ab @N0b þ þ pb ¼ 0 @a @b

ð7Þ



N0b R

þ pc ¼ 0

ð8Þ

in which pa, pb and pc are the components of the inertia load exerted on the shell (see Fig. 2). It follows from Fig. 3 that

pa ¼ 0

ð9Þ

b pb ¼ p sin R pc ¼ p cos

ð10Þ b R

ð11Þ

Substituting Eq. (11) into Eq. (8) yields

N0b ¼ pR cos

b R

ð12Þ

Integrating Eq. (7) with respect to a and taking into account Eqs. (10) and (12) the following equation is derived: Fig. 1. Clamped–clamped cylindrical shell subjected to the transverse inertia loading.

b N0a ¼ 2pa sin þ fab ðbÞ R

ð13Þ

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A.V. Lopatin, E.V. Morozov / Composite Structures 95 (2013) 471–478

!

2

pl B12 b pR cos  R 6 B22 R

fa ðbÞ ¼

ð22Þ

Taking this into account, Eq. (16) is transformed as follows

"

2

p l Na ¼  la þ a2 R 6 0

!

# B12 b  pR cos R B22

ð23Þ

Thus, all the membrane pre-buckling stress resultants are determined. The respective equations, Eqs. (12), (15) and (23) can be presented in the following form:

b N0a ¼ Ng a ðaÞ cos ; R

b N0b ¼ N cos ; R

N0ab ¼ Ng ab ðaÞ sin

b R

ð24Þ

in which N = pR and

g a ðaÞ ¼

in which fab(b) is unknown function. Since N 0ab ¼ 0 for a = l/2, then it follows from Eq. (13) that

Nab

b R

ð14Þ

b ¼ p sin ðl  2aÞ R

ð15Þ

Integrating Eq. (6) with respect to a and taking into account Eqs. (9) and (15) the following relationship is derived:

p b N0a ¼  cos ðla  a2 Þ þ fa ðbÞ R R

e0a ¼

g ab ðaÞ ¼

l a 12 R l

N0b ¼ B21 e0a þ B22 e0b

@u0 @a

þ

B22 @w D21 þ 2D33 @ 3 w D22 @ 3 w  ¼0  @ a2 @b R @b3 R @b R



B21 @u B22 @ v D22 @ 3 v D12 þ 2D33 @ 3 v  þ þ R @ a R @b R2 @b3 R @ a2 @b

@4w @4w @4w  2ðD12 þ 2D33 Þ  D22 4 2 4 @a @ a2 @b @b R " !# ! 2 2 g b 1 @v @ w @ w b @2w ab @ v ¼0  þ N cos þ2 þ g a 2  2 þ sin R R @b @a R @ a@b R @a @b

ð16Þ

ð17Þ ð18Þ

B22 2

w  D11

ð26Þ Hence, the buckling boundary problem for the shell under consideration is reduced to Eq. (26) supplemented with boundary conditions, Eq. (5).

in which e0a ; e0b are the longitudinal and circumferential membrane pre-buckling strains and u0 is the pre-buckling displacement of the points of middle surface along the a-axis. Solving Eq. (17) for e0a yields

e0a ¼

 1 B22 N0a  B12 N0b B

ð19Þ

where B ¼ B11 B22  B212 . Substituting Eqs. (12), (16) and (19) into Eq. (18) and integrating the resulting equation with respect to a, the following formula for the longitudinal displacement is derived:

u0 ¼

     p b a2 a3 b l  B22  cos þ fa ðbÞa þ B12 pRa cos R R R 2 3 B

1

þ f ðbÞ

ð20Þ

According to the boundary condition, the displacement u0 = 0 at a = 0. Then, it follows from Eq. (20) that f(b) = 0. The function fa(b) in Eq. (16) can be found using the following apparent condition:

u0 ða ¼ l=2Þ ¼ 0

ð25Þ

@2u @2u @ 2 v B12 @ 2 w þ B þ ðB þ B Þ ¼0 þ 33 12 33 @ a2 @ a@b R @ a @b2     @2u D33 @ 2 v D22 @ 2 v ðB21 þ B33 Þ þ B þ þ B33 þ 2 22 @ a@b @ a2 R R2 @b2



In order to find the unknown function fa(b), the following constitutive and strain–displacement relations of the membrane theory of shell are employed:

N0a ¼ B11 e0a þ B12 e0b ;

2

B11

As a result, Eq. (13) takes the following form: 0

2

Substituting constitutive equations, Eq. (3), strain–displacement relationships, Eq. (4) and expressions for the pre-buckling stressresultants, Eq. (24) into the linearised buckling equations, Eq. (2), the latter can be expressed in terms of displacements u, v and deflection w as follows

Fig. 3. Components of the inertia load exerted on the shell.

fab ðbÞ ¼ pl sin

  1 a a2 B12  þ 2  ; B22 R 6 l l l

ð21Þ

Applying this condition to Eq. (20) and solving the resulting equation for fa(b) (taking into account that f(b) = 0) the following equation is derived:

Fig. 4. Clamped–clamped beam function (a) and its first derivative (b).

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3. Solution procedure

According to Galerkin method, the orthogonality conditions for the errors, Eq. (30) and the basis functions have the form

The problem can be solved using Galerkin method which is, in this case, applied in two stages, i.e. separately for each coordinate. Selection of the approximating functions, as per Galerkin procedure, is based on some assumptions with regard to the buckling mode shapes of the shell loaded with the inertia forces. It follows from the analysis of Eqs. (24) and (25) that the top part of the shell is compressed in both directions, axial and circumferential. Taking such distribution of the internal pre-buckling membrane forces (stress resultants) into account, assume that the buckling mode shape is characterised with the formation of the buckling lobes oriented in the axial direction, i.e. along the a-axis. In this case, the deflections u, v and deflection w vary in the axial direction substantially slower than in the hoop direction and can be approximated with the first mode clamped–clamped beam functions and their derivatives at the first stage of the application of Galerkin method as follows [11]

Z

u¼

dXðaÞ UðbÞ; da

l

Ra ða; bÞ

0

Z

Z

dX da ¼ 0; da

l

Rb ða; bÞXda ¼ 0;

0

l

Rc ða; bÞXda ¼ 0

ð31Þ

0

Substituting Eq. (30) into Eq. (31) and integrating the resulting equations allowing for the formulas [12] Z l  2 Z l 2 3 d X f dX f dX d X k4 da ¼  ; da ¼ ; da ¼  3 ð32Þ 2 3 da l da l 0 0 0 0 da da l Z l Z l 2 dX l d X n g ab X da ¼ ; g a X 2 da ¼ da R da l 0 0 Z

l

X 2 da ¼ l;

Z

l

X

in which 2

v ¼ XðaÞVðbÞ;

w ¼ XðaÞWðbÞ

ð27Þ

where

f ¼ rkðrk  2Þ;

n¼

l 2rk  3 B12 þ f; 6 B33 R2

ð33Þ

the following system of ordinary differential equations is derived:

  ka ka ka ka XðaÞ ¼ cosh  cos  r sinh  sin l l l l

ð28Þ

and k = 4.73004074, r = 0.982502215 [12]. The corresponding first derivative of the function X(a) is given by

   dXðaÞ k ka ka ka ka ¼ sinh þ sin  r cosh  cos da l l l l l

ð29Þ

l dX are shown in Fig. 4a and b. It k da follows from Eqs. (28) and (29) that X = 0, dX/da = 0 for a = 0 and a = l. In turn, it follows from Eq. (27) that the selected basis functions of coordinate a satisfy the boundary conditions given by Eq. (5). Substituing Eq. (27) into Eq. (26) and taking into account that The graphs of functions X(a) and

4

d X k4 ¼ 4 X yields the following expressions for the errors (as per da4 l Galerkin procedure): 3

k4

2

fd U f dV B12 f þ W ¼0 þ ðB12 þ B33 Þ 3 l db2 l db R l l     2 f dU D33 f D22 d V  B33 þ 2 V þ B22 þ 2 l 2  ðB21 þ B33 Þ l db l db R R   3 l D21 þ 2D33 f dW l d W  D22 þ B22 þ ¼0 R l db R db3 R   3 B21 f l D21 þ 2D33 f dV l d V U  B22 þ þ D22 R l db R db3 R l R ! 2 4 B22 k4 fd W d W  þ D11 4 lW þ 2ðD12 þ 2D33 Þ  D22 l 2 2 l db db4 R l ! "  # 2 b l dV n d W b l V dW þ sin þ W l  þ 2 ¼0 þ N cos R R db l RR R db db2  B11

U þ B33

ð34Þ

2

d X dX d U dX dV B12 dX U þ B33 þ ðB12 þ B33 Þ W þ da3 da db2 da db R da   2   2 2 d X dU D33 d X D22 d V Rb ða; bÞ ¼ ðB21 þ B33 Þ 2 V þ B22 þ 2 X 2 þ B33 þ 2 2 a da db d db R R ! 2 3 B22 D21 þ 2D33 d X dW D22 d W X X þ  R R R da2 db db3 ! 2 2 3 B21 d X B22 D12 þ 2D33 d X dV D22 d V X  X 3 Rc ða; bÞ ¼  U  þ R da2 R R R da2 db db ! 2 2 4 B22 k4 d Xd W d W  þ D  D22 X XW  2ðD12 þ 2D33 Þ 2 11 2 4 2 4 a d db db R l " !  # 2 2 g ab dX b X dV d X d W b dX dW þ N cos V þ 2g þ ga 2 W  X  þ sin a b R da R R db da R da db db2 Ra ða; bÞ ¼ B11

ð30Þ

This system of equations is solved at the second stage of application of Galerkin method. According to this approach, the functions U(b), V(b) and W(b) are presented in the following trigonometric series

U¼

X U n cos kn b;

W¼

n X W n cos kn b

V¼

X V n sin kn b; n

ð35Þ

n

in which Un, Vn and Wn are unknown variables, kn = n/R, and n is the number of buckling waves in the hoop direction of the shell. Substituting series, Eq. (35) into Eq. (34) and replacing the subscript n with n1 the following equations for the errors are derived:

" ! # X k4 f f B12 f W n1 cos kn1 b  B11 3 þ B33 k2n1 U n1 þ ðB22 þ B33 Þ kn1 V n1 þ l l R l l n1         X f D33 f D22 l D21 þ 2D33 f l þ B22 þ 2 lk2n1 V n1  B22 þ kn1 þ D22 k3n1 W n1 sin kn1 b ðB22 þ B33 Þ kn1 U n1  B33 þ 2 Pb ðbÞ ¼ l l R l R R R R n1 " ! # )    X B21 f l D12 þ 2D33 f l B22 k4 f 2 4 Pc ðbÞ ¼ l þ 2ðD W cos kn1 b U n1  kn1 B22 þ þ D22 k2n1 V n1  k þ D þ D k þ 2D Þ 11 22 12 33 n n1 1 4 R l R l n1 R l R R2 l n1     X l  b 1 b n b l b þN kn1 cos cos kn1 b  sin sin kn1 b V n1 þ þ lk2n1 cos cos kn1 b  2 kn1 sin sin kn1 b W n1 R R R R l R R R n

Pa ðbÞ ¼

1

ð36Þ

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A.V. Lopatin, E.V. Morozov / Composite Structures 95 (2013) 471–478

The orthogonality conditions for the errors, Eq. (36) and the corresponding basis functions are given by

Z 2pR

Pa ðbÞ cos kn bdb ¼ 0;

Z 2pR

0

Z 2pR

Pb ðbÞ sin kn bdb ¼ 0; ð37Þ

0

Substituting the errors, Eq. (36) into Eq. (37) and integrating the resulting equations with respect to b allowing for the following relationships 

Z 2pR

b pR 0; coskn bcos cos kn1 bdb ¼ R 2 1; 0  Z 2pR b pR 0; coskn bsin sin kn1 bdb ¼ R 2 1; 0

if n þ 1 – n1



 þ

0; if n  1 – n1



if n þ 1 ¼ n1 1; if n  1 ¼ n1    0; if n  1 – n1 if n þ 1 – n1 þ if n þ 1 ¼ n1 1; if n  1 ¼ n1

ð38Þ in which kn ¼ n=R; kn1 ¼ n1 =R, the following homogeneous system of linear equations is derived after some rearrangements:

! k4 B11 þ B33 fn2 s U n  ðB12 þ B33 ÞfnV n  B12 fW n ¼ 0 s      D33 f D22  ðB21 þ B33 ÞfnU n þ B33 þ 2 þ B22 þ 2 n2 s V n s R R     D22 2 D21 þ 2D33 f þ n B22 þ 2 n s þ Wn ¼ 0 s R R2     D22 D12 þ 2D33 f Vn  B21 fU n þ n B22 þ 2 n2 s þ 2 s R R " ! # D11 k4 D22 D21 þ 2D33 f 2 n Wn þ B22 þ 2 4 þ 2 n4 s þ 2 s R s R R2 s  N ½nðV nþ1 þ V n1 Þ þ r n ðW nþ1 þ W n1 Þ ¼ 0 2 2

ðnÞ

ðnÞ

ðnÞ

ðnÞ A31 U n

ðnÞ A32 V n

ðnÞ

ð39Þ

2

ðnÞ

ðnÞ

A21 U n þ A22 V n þ A23 W n ¼ 0 ð40Þ

ðnÞ A33 W n

þ þ s  g ½nðV nþ1 þ V n1 Þ þ r n ðW nþ1 þ W n1 Þ ¼ 0 2 

ð41Þ

ðnÞ

ðnÞ

A13

ðnÞ A23

ðnÞ

A33

B11 R2 k4 B33 R2 2 fn s; þ D22 s D22

ðB12 þ B33 ÞR2 ðnÞ ðnÞ A12 ¼ A21 ¼ fn D22 ! ! B12 R2 B33 R2 D33 f B22 R2 ðnÞ ðnÞ þ ¼ A31 ¼ f; A22 ¼ þ þ 1 n2 s D22 D22 D22 s D22 " # ! 2 B22 R D21 þ 2D33 f ðnÞ ¼ A32 ¼ n þ n2 s þ s D22 D22 ! 2 4 B22 R D11 k D12 þ 2D33 2 f ð42Þ ¼ þ þ n4 s þ 2 n s D22 D22 s4 D22

Parameter g in Eq. (41) is the dimensionless buckling coefficient which is defined as

g¼

NR2 D22

ð43Þ

where

V n ¼ FnW n

ðnÞ

Fn ¼

ðnÞ

ðnÞ

ðnÞ

A12 A13  A11 A23  2 ðnÞ ðnÞ ðnÞ A11 A22  A12

ð45Þ

Hnn W n  gðQ n;n1 W n;n1 þ Q n;nþ1 W n;nþ1 Þ ¼ 0

ð46Þ

in which ðnÞ

ðnÞ

ðnÞ

Hnn ¼ A33  A13 C n þ A23 F n Q n;n1 ¼

s ðnF n1 þ r n Þ; 2

Q n;nþ1 ¼

ð47Þ

s ðnF nþ1 þ r n Þ 2

Using truncated series, Eq. (35) (retaining only k terms), the following homogeneous system of k linear equations in k unknowns is constructed with the aid of Eq. (46):

HW  gQW ¼ 0

ð48Þ

where

Q¼

8 > > > > > Q 21 > > > > <

9 > > > > > > > > > =

H22 H33 H44 ...: Hkk

Q 23 Q 32

ð49Þ

> > > > > > > > > ;

9 > > > > > > > > > =

Q 12

> > > > > > > > > :

Q 34 Q 43

...: ...: Q k;k1

> > > > > Q k1;k > > > > ;

Hence, the buckling problem under consideration is reduced to the generalised eigenvalue problem specified by Eq. (48). Solution of this problem yields the critical buckling coefficient gcr as the minimum eigenvalue:

acr ¼ ðgnc Þcr ¼ gcr

D22

ð51Þ

Bq R3

4. Numerical analysis Consider examples of calculations of the critical accelerations for shells made from different materials. In the first example, the orthotropic composite shell of thickness h is analysed. The material properties are characterised by the moduli of elasticity Ea, Eb and Gab and Poisson’s ratios mab and mba. The membrane and bending stiffnesses of the shell are

B12 ¼ Ea mab h; 3

ð44Þ

ð50Þ

The required accuracy of calculation of the coefficient gcr is achieved by solving the problem and comparing the results for different values of k. Once the buckling coefficient gcr is determined, the critical acceleration can be calculated using Eqs. (1) and (43) taking into account that N = pR as follows

B11 ¼ Ea h;

Solving equations, Eq. (40) for U n and Vn yields

Un ¼ Cn W n ;

ðnÞ

gcr ¼ minðg1 ; g2 ; g3 ; g4 ; . . . ; gk Þ

in which the dimensionless coefficients are given by

A11 ¼

ðnÞ

8 8 9 W1 > H11 > > > > > > > > > > > > > > W2 > > > > > > > > > > > < W3 > < = W¼ H¼ > > > W4 > > > > > > > > > > > > > > > >...:> > > > > > > > : : ; Wk

Here, U n ¼ U n =l; s ¼ l=R and rn = n/s + n  1. Multiplying Eq. (39) by R2/D22 the latter are transformed into the following form:

A11 U n  A12 V n  A13 W n ¼ 0;

ðnÞ

A13 A22  A12 A23  2 ; ðnÞ ðnÞ ðnÞ A11 A22  A12

Substituting for U n and Vn in Eq. (41) their expressions, Eq. (44), the following equation is derived:

0

Pc ðbÞ cos kn bdb ¼ 0

ðnÞ

Cn ¼

D11 ¼ Ea

h ; 12

B22 ¼ Eb h;

B33 ¼ Gab h

3

D12 ¼ Ea mab

h ; 12

3

D22 ¼ Eb

h ; 12

3

D33 ¼ Gab

h 12 ð52Þ

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A.V. Lopatin, E.V. Morozov / Composite Structures 95 (2013) 471–478

where Ea ¼ Ea =ð1  mab mba Þ and Eb ¼ Eb =ð1  mab mba Þ. Substituting Eq. ðnÞ (52) into Eq. (42), the coefficients Aij ði; j ¼ 1; 2; 3Þ are calculated as follows ðnÞ

A11 ¼ 12t 2 ðnÞ

! Ea k4 Gab 2 fn s ; þ Eb s Eb

ðnÞ

A33

mba þ !

Gab f þ n2 s Eb s " ! # Gab f 2 2 ¼ n ð12t þ n Þs þ mba þ Eb s ! ! 4 Ea k Gab 2 f 2 4 n ¼ 12t þ þ n s þ 2 mba þ 2 s Eb s4 Eb

A12 ¼ 12t 2 mba f; ðnÞ A23

ðnÞ

A12 ¼ 12t2

Gab Eb

! fn

gcr Eb 12t3 Bq

B12 ¼ Emh;

B11 ¼ B22 ¼ Eh;

ðnÞ

A22 ¼ ð12t 2 þ 1Þ

3

ð53Þ

D11 ¼ D22 ¼ E

h ; 12

1m Eh 2 3 1m h E D33 ¼ 2 12

B33 ¼ 3

D12 ¼ Em

h ; 12

ð55Þ

where E ¼ E=ð1  m2 Þ; E is Young’s modulus and m is Poisson’s ratio. ðnÞ The coefficients Aij ði; j ¼ 1; 2; 3Þ in Eqs. (40) and (41) are given by ðnÞ A11

where t = R/h. The parameter n in Eq. (33) is calculated for an orthotropic shell as n = s2(2rk  3)/6 + mba f. The mass per unit area is Bq = qh, where q is material’s density. Substituting D22 in Eq. (51) with its expression as per Eq. (52), the critical acceleration can be calculated as follows

acr ¼

critical acceleration can be increased if the shell is reinforced with the stiffening rings spread along the length of the cylinder. Buckling of the isotropic shell has been considered in the second example. In this case, the membrane and bending stiffnesses are calculated as follows

ð54Þ

The shell is made of glass fibre reinforced polymer with the following properties: Ea = 14 GPa, Eb = 14 GPa, Gab = 1.8 GPa, mba = 0.08, mab = 0. 08, q = 1850 kg/m3. The critical acceleration has been calculated for the shells having the same diameter: 1 m (R = 0.5 m). However, the length, l and thickness of the wall, h were different: l = 1 m, 2 m, 5 m and 10 m and h = 0.5 mm and 1 mm. The results of calculations of acr are presented in Table 1. The critical buckling coefficient gcr has been calculated for k = 100. The results of calculations have been verified by comparison with the finite-element solutions. The problem has been solved using COSMOS/M [13]. The shells were modelled using 10  10 mm shell elements SHELL4L. So, for example the finite-element model for the shell with l = 1 m was built of 31400 elements. The computational time ranged from 1 to 4 min depending on the length of the shell. The results of calculations of the critical accelerations aFEM for the selected values of l and h are shown in Table 2. cr As can be seen, the differences between acr and aFEM do not exceed cr 5.83% (l = 10 m, h = 1 mm). It follows from the Table 2 that the shorter the shell, the less the difference between the acceleration values calculated by the two methods. The buckling mode shapes for the shells with R = 0.5 m, h = 1 mm and l = 1 m and 10 m are shown in Figs. 5 and 6, respectively. As can be seen, these shapes coincide completely with those originally assumed in the solution procedure (see Section 3). Thus, the assumptions made in this work with regard to the buckling modes proved to be right. It follows from Figs. 5 and 6 that the length of the shell affects considerably the number and size of the buckling lobes formed in the hoop direction (axis b). The short shell (l = 1 m) buckles in the upper part of the cylinder surface (see Fig. 5) with formation of a few small buckling waves. The deflection amplitude of these waves decreases with the increase in distance from the top of the shell (b = 0) in the hoop direction. Note that rather similar buckling modes have been observed for the shells with the lengths l = 2 m and 5 m, however the number of buckling lobes has been reduced for the longer shells. For example, it could be seen in Fig. 6 that the long shell (l = 10 m) buckles with formation of fewer shallow buckling waves spreading virtually all over the circumference of the cross-section. Note that based on the observed above buckling modes some recommendations could be made with regard to the shell design which would improve its load carrying ability. In particular, the

¼ 12t

2

! k4 1  m 2 þ fn s ; s 2

ðnÞ

A13 ¼ 12t2 mf; ðnÞ A23

ðnÞ

ðnÞ

A22 ¼ ð12t 2 þ 1Þ

  f ¼ n ð12t 2 þ n2 Þ s þ ; s

1þm fn 2  f þ n2 s s

A12 ¼ 12t 2

ðnÞ A33

 1m 2 ¼

! k4 f 4 12t þ 4 þ n s þ 2n2 s s 2

ð56Þ The parameter n in Eq. (33) is calculated for isotropic shell as n = s2(2rk  3)/6 + mf. Substituting Eb ¼ E into Eq. (54), the following equation for the critical acceleration can be derived:

acr ¼ wcr

E Bq

ð57Þ

in which

wcr ¼

gcr

ð58Þ

12t 3

For given value of Poisson’s ratio, the critical coefficient wcr depends only on two geometric parameters, i.e. s = l/R and t = R/h. The effects of these parameters have been studied for 2 6 s 6 10 and 500 6 t 6 2500 and m = 0.3. The generalised eigenvalue problem specified by Eq. (48) has been solved for given pair of s and t and the corresponding critical buckling coefficient gcr was found according to Eq. (50). Analyses were performed for k = 100. Then, the coefficient wcr was calculated using Eq. (58). The function wcr(s, t) is shown in Fig. 7. It follows from the data presented that the value of wcr reduces significantly with the increase in the values of parameters s and t. The following approximate analytical formula for wcr has been found using CurveExpert [14]:

wcr ¼

b sc t d

ð59Þ

in which b = 1.695308, c = 0.895782 and d = 2.537325. Using Eqs. (57) and (59), the dimensions of the isotropic cylindrical shell corresponding to a specified critical acceleration can be identified. Substituting Eq. (59) into Eq. (57) and taking into account that Bq = qh, s = l/R and t = R/h the following equation is derived: d1

acr ¼

E bh q lc Rdc

ð60Þ

Solving this equation for l yields Table 1 Critical accelerations acr (m/s2) for the orthotropic shells with different length, l and thickness, h. h (m)

l (m) 1

2

5

10

0.0005 0.001

319.03 892.93

175.06 500.87

79.14 232.37

44.17 131.89

A.V. Lopatin, E.V. Morozov / Composite Structures 95 (2013) 471–478

477

Table 2 2 Critical accelerations aFEM cr ðm=s Þ for the orthotropic shells with different length, l and thickness, h. h (m)

0.0005 0.001

l(m) 1

2

5

10

317.09 881.08

176.49 499.03

78.20 225.65

42.09 124.63

Fig. 6. Buckling mode of the orthotropic cylindrical shell (l = 10 m).

Fig. 7. Function wcr(s, t) for the isotropic shell.

Fig. 5. Buckling mode of the orthotropic cylindrical shell (l = 1 m).

l¼

!1 d1 c E b h q a~cr Rdc

ð61Þ

~cr where a is a given critical acceleration. If ~cr ¼ 500 m=s2 ; E ¼ 210 GPa; m ¼ 0:3; R ¼ 0:5 m and h = 0.001 m, a then according to Eq. (61), the required length of the shell is found to be l = 9.69 m. Similarly, the thickness, h and radius of the shell, R could be found using Eq. (60) for given l and R or l and h, respectively. 5. Conclusions The buckling problem has been solved for the composite orthotropic cylindrical shell having fully clamped ends and subjected to

the transverse inertia load. The pre-buckling stress state is considered to be inhomogeneous and the corresponding pre-buckling membrane stress resultants have been found using the membrane theory of cylindrical shells. The governing buckling equations with variable coefficients have been solved using Galerkin method. The shell displacements in the axial direction have been approximated with the first mode clamped–clamped functions in accordance with the assumed buckling mode. The corresponding hoop displacements were represented by the trigonometric series. Solving the resulting generalised eigenvalue problem, the dimensionless critical buckling coefficient has been found and the corresponding critical acceleration calculated. The critical accelerations have been calculated for various combinations of geometric parameters of composite glass fibre reinforced orthotropic shells. Results of calculations have been validated by comparison with the computations based on

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finite-element modelling and analysis. The finite-element analysis has also confirmed the correctness of assumptions made with regard to the buckling mode shapes in the proposed solution procedure. In addition, the analytical formulas have been obtained enabling the design analyses of isotropic cylindrical shells to be performed for the loading case under consideration. The new non-classical buckling mode has been identified for the cylindrical shell loaded with transverse inertia forces. Major features of this mode are specified by the way the shell is supported at the ends and the character of loading. The results obtained in this work can find their applications when designing and analysing cylindrical shells subjected to body forces. References [1] Flügge W. Stresses in shells. 2nd ed. Berlin: Springer Verlag; 1973. [2] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1963.

[3] Brush DO, Almroth BO. Buckling of bars, plates and shell. New York: McGraw Hill; 1975. [4] Yamaki N. Elastic stability of circular cylindrical shells. Amsterdam: Elsevier; 1984. [5] Volmir AS. Stability of deformable systems. Moscow: State Publishing House (Nauka); 1967 [in Russian]. [6] Alfutov NA. Stability of elastic structures. Berlin: Springer Verlag; 2010. [7] Jones RM. Buckling of bars, plates and shells. Blacksburg, Virginia: Bull Ridge Publishing; 2006. [8] Lim CW, Ma YF, Kitipornchai S, Wang CM, Yuen R. Buckling of vertical cylindrical shells under combined end pressure and body force. J Eng Mech 2003:876–84. [9] Lim CW, Ma YF. Computational p-element method on the effects of thickness and length on self-weight buckling of thin cylindrical shells via various shell theories. Comput Mech 2003;31:400–8. [10] Vasiliev VV. Mechanics of composite structures. Washington, DC: Taylor & Francis; 1993. [11] Leissa AW. Vibration of shells. Acoustical Society of America; 1993. [12] Blevins RD. Formulas for natural frequency and mode shape. Malabar, FL: Krieger Publishing Company; 2001. [13] COSMOS/M user guide. Structural Research & Analysis Corporation; 2003. [14] Hyams DG. CurveExpert Professional Documentation Release 1.5.0; 2011.