Indian Journal of Engineering & Materials Sciences Vol. 10, October 2003, pp. 365-370

Buckling of an orthotropic cylindrical thin shell with continuously varying thickness under a dynamic loading

Abdullah H Sofiyeva*, Erol M Keskina, Hakan Erdemb & Zihni Zerinb

' Department of Civil Engineering, SUleyman Demirel University, Isparta, Turkey

bDepartment of Civil Engineering, Ondokuz MaYls University, Samsun, Turkey

Received 7 January 2003; accepted 21 August 2003

The buckling of an orthotropic composite cylindrical she ll with variable thickness, subjected to a dynamic loading, is reported here. At first, the fundamental relations and Donnell type dynamic buckling equation of an orthotropic cylindrical shell with variable thickness have been obtained. Then, employing Galerkin's method, these equations have been reduced to a time dependent differential equation with variable coefficients. Finally, for different initial conditions and approximation functions, applying the Ritz type variational method, analytical expression has been found for the dynamic factor. Using these results , the effect of the variations of the power of time in the external pressure expression, the loading parameter and the ratios of the Young's moduli on the dynamic factor are studied numerically for the case when the ihickness of the cylin­drical shell varies as a power and exponential functions. It has been observed that these effects change the dynamic factor of the problem in the heading appreciably .

Shells are one of the necessary structural components and commonly found in a variety of engineering ap­plications, such as the military , aerospace, turbo ma­chinery and ship building industries. In recent years, new composite materials made of two or more com­ponents are used in many fields of industry. Shells composed of composite materials constitute different structures having orthotropic characteristics I.

There are numerous methods to determine the static critical loads, consistent with experimental results for uniform orthotropic composite thin shells under dif­ferent loads with different boundary conditions. There are some reports about orthotropic composite thin shells with variable thickness due to the difficulties in their production and theoretical analysis2

.4

. None-the­less , it is highly probable that this type of structural parts will be used a lot in the future due to the advan­tages of their light weights and small volume and the progress in their fabrication methods. In recent years, numerous research works have been reported the buckling and vibration of shells with variable thick­ness5. 11

.

The effect of the variation of thickness on the dy­namic factor has not been studied appreciably. The solution of a dynamic problem is reduced to the de­termination of the dynamic factor for certain loading

*Author for correspondence

cases. The dynamic factor can be found, using differ­ent methods, depending on the manner in which the loading is applied, particularly on the loading pa­rameter I3

.16

.

The aim of the present study is to investigate the buckling of an orthotropic composite cylindrical shell with continuously varying thickness subjected to a uniform external pressure which is a power function of time, for different initial conditions and approxi­mation function using the Ritz type variational method.

Theoretical

Equations of motion

Consider a circular cylindrical thin shell of me­dium length composed of orthotropic composite mate­rials , with immovable simple supports at the ends, having length L, radius r and thickness h. The right­handed system of coordinates is selected in such a

way that the origin is on the middle surface, the OX 3

axis is perpendicular to the middle surface of the

plane, positive inwards, and the OX 1 and OX 2 axes

are in the axial and circumferential directions, respec­tively (Fig. 1). h is a continuous and second order dif­ferentiable function with respect to in the axial direc­tion.

366 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

I l I

p~)11111111111 0

x,

p~)11111111111 II Fig. 1 - Cylindrical shell of variable thi ck ness h subjected to a dynam ic loadi ng C", , exponenti al; - - -, linear; _. - , para-

bolic; _ .. -, cubi c.)

Taking account of the radial inertia forces, the modified Donnell type basic equations of cylindrical shell are found as I 3

.16

:

II~ I ' .v, +21121 2 ' X,X1 +l1lz2 ' X1 X2 +n22 / r + 11~1 U 'XIX,

+2n~2U'X'X1 +n~2U'X2X, = phu'll (1)

(2)

where mil ' I1lz2 and m l 2 are axial , circumferential

and twisting moments, respectively, n22 is c ircumfer-

. I I' 0 0 dO . I entia lorces, nil' n 22 an n l 2 are Incrementa stress

resultants about fundamental state, u is the displace­ment of the middle surface of the shell in the normal

direction, positive inwards, e~1 and e~2 are the normal

strains in the directions of the axial and circumferen­

ti al axes, respectively, and e~2 is the shear strain (all

on the same surface), (,) denoting differentiation with respect to, p is material s density and t the ti me.

The well-known force and moment resu ltants are expressed by:

[(1111 , /l22 , n I2 ), (11211' m 22 , m I2 )]

11 12

= f [1,x3 ] [0"11 ' 0"22 ' 0"12] dx3 - 11 12

. .. (3)

The relation between the forces and the Airy stress

function <p = cp/ ho is given by:

... (4)

where ho is the nominal thickness of the shell.

But, the Hooke's Law for an orthotropic material is 17:

... (5)

Therefore,

Q = Ell En II ' Q22 =----,

1- 111 2 1121 1- J.1J 2 1121

E"2 E"2 Q _ II 22 Q Q Q

33 -1+ /1 "2 /1" 2' 12 =1121 I I =111 2 22 .... 2 1 .... 12

... (6)

in which Ell and E22 are the Young's moduli in the

directions of the OX I and OX 2 axes , 1112' 1121 are the

Poisson' s ratios of the orthotropic material and

1121 Ell = 111 2 E 22 .

The shell is subjected to an external pressure vary­ing as a power function of time:

... (7)

where p is the loading parameter and q is the power

of time in the external pressure expression, the last one being equal to or greater than unity .

After substituting Eqs (3-7) into relations (1) and

(2), and using dimensionless coordinates ~I = XI / L,

~2 = x2 / r , we get the system of matrix equations:

(8)

where the differential operators L"II. (m, n = 1 ,2) ap­pearing in Eq. (8) are defined according to:

12(1-11121121) L2 a2

Ell rh3 a~12 ... (10)

SOFIYEV el at.: BUCKLING OF AN ORTHOTROPIC CYLINDRICAL THIN SHELL 367

(11)

. . . (12)

Solution of eigenvalue problem

Assuming the shell to be simply supported along the peripheries of both bases, the solution of the sys­tem of Eq. (8) is sought as follows 13:

cP = I Icpij (t )sin jn~1 sin i~2' • J

u=IIuij(t)sin jn~1 sini~2 ... (13)

• J

where j is the number half-waves along the shell

length at buckling in the axial direction, i denotes the number of waves circumferential direction during

buckling and CPij (t) and uij (t) are time dependent

amplitudes. Substituting expressions (13) in Eq. (8) and then applying Galerkin ' s method in the ranges

o ~ ~I ~ 1 and 0 ~ ~2 ::; 2n in obtained equation set,

the terms that are included by expression (13) are solved. One of these terms is selected and eliminating

CPij (t) from the equations, thus obtained, and remem-

bering that when the half wave number j is equal to

one, the wave number i for a shell of medium length

satisfies the inequality i4 »8 4, the following equa­

tion is obtained I 1.12:

where 8 = n r / L, t = r ter , in which t cr is the critical

time and the dimensionless time parameter 't satisfies 0::; r::; 1, and the following definitions apply:

I I

81 = I h3 (~I )sin2 n~,d~" 82 = I h-' (~I )sin

2 n~,d~" o 0

I

PI = pI h(~,) sin2n~,d~1 ... (15) o

Eq. (14) is solved for two initial conditions . (i) In first approximation, the function satisfying initial

conditions Uo (0) = 0, Uo 't (0) = 0 is in the form:

.. . (16)

(ii) Furthermore, the curve (uo, -c) has the maximum

when r = 1, so in first approximation the function

satisfying initial conditions uo(O)=O, uO't(l)=O is

in the form:

... (17)

h LI ' l' d II 12 were, 'ij IS amp Itu e ..

Applying the Ritz type variational method to dif­ferential Eq. (14) and minimizing characteristic equa­tion according to the wave number i, an equation is

obtained . After solving this equation and after mathematical operations, the following expression for d . f . f d" 12 . ynamlc actor IS oun ':

1[4 P 21q r (5+3q)l q8(I+q)/(2q) ]2q

K = 1. 1398 x I Po 2 d 2(I+q)1 q ('+q) /(20)

8 Ell

[

3(1 - J.112J.121 ) ](3+q

) [~q ~2q )}"(4q

+4)

E2281

A;q ... (18)

where Au, ~ and A2 are defined as:

Ao = f[uo (-c)]2 d-c, Al = f[uo.t (-c)p d-c, o 0 IT

Az = 2 If 1]q Uo ''1 (1]) uo (1]) d1]dr ... (19) 00

When h = constant from the expressions (18) for a uniform orthotropic composite cylindrical shell, the appropriate formula is found as a special case.

Results and Discussion The case of an orthotropic composite cylindrical

thin shell in which the thickness changes as a power

368 INDIAN J. ENG. MATER. SCI., OCTOBER 2003

6,5 1'. I p=150 (MPafs"). q=1 I . . '.' . . . . ~ . ""-

....... , .. 6

u E 5,5 l! e: 5 ~

'., "'--.., ~'-~-.....

4 ,5

0,6

-

0,7

- - +- - in. - ... -cub.

~"'" '" ... ..~

--....:":::! ~ .. ... ~~.,

O,S!

-'p&". - ... -exp.

p=150 (MPals) , q=1 , h/h~=O.6

4+-______ ~------~1--------25 50 75 100

En/En

I --+- - in. - ' p&". . - 4- -!!DIp. --00.

Fig. 2 - Variations of dynamic factor with the ratio h, 1 h2 and E" I E22

Table. I - Variations of the dynamic factor with power of time q and loading parameter p

p MPals

110 (-r) = Ae50Tr[52/51 - r]

110 (0) = 0 , 1I0 ' T (I) = 0 ,

Linear (hllh2=0.75)

Uniform (hllh2=1)

110 (r) = Ae5DTr 2 [53/52 - r 1

IIO(O) = O,IIO'T(O) = O,

Linear (h/h2=0.75 )

Uniform (h/h2=I )

100 150 200 100 150 200 100 150 200 Kd

100 150 200 q Kd Kd

4.57 5.60 6.47 3.77 4.62

2 0.55 0.63 0.69 0.47 0.54

3 0.19 0.21 0.23 0.17 0.19

and exponential functions of time has been considered by Irie et al. 4

• The thickness at the two ends being h, and h2 , this case can be expressed in the following manner:

(20)

. .. (21)

where k = 1,2,3 ... , correspond to the cases of linear, parabolic, cubic etc. variations, respectively (Fig. I).

Numerical computations have been carried out for the following data used in the computations, G h· / . I . '7 ' 8 rap Ite epoxy materIa propertIes '

E" = 1.724 x 105 MPa, E22 =7.79x103 MPa,

1112 = 0.35, 112, = 0.016, P = 1530 kg/m 3, shell

properties '2.15 ~ =8 x lO-4 m, r=9xlO-2 m,

L = 0.2 m. Using above data, numerical results found for various loading parameters p and power of timeq

values have been presented in graphical and tabular forms. The approximation function (17) is used in

5.33 4.62 5.65 6.53 3.81

0.60 0.56 0.64 0.70 0.48

0.20 0.20 0.22 0.23 0.17

4.66 5.38

0.55 0.60

0.19 0.21

Fig. 2. But in Table 1, all of two approximation func­tions (l 6) and (17) are used.

Fig. 2 shows the variations of the dynamic factor with the ratio h, / ~ and Ell / E22 . In orthotropic com­

posite cylindrical thin shells with thickness varying as a linear, parabolic, cubic or exponential function in the axial direction, a decrease in the ratio I"" / ~

causes an increase in the dynamic factor. When the foregoing ratio becomes one, the latter quantities take their values for a shell of uniform thickness. When thickness varies exponentially, the highest effect to the dynamic factor is 40.7%. When h/h2 = 0.6 and

with an increase of the ratio Ell / E22 , the values of the

dynamic factor increase. In the case at which the thickness ~hanges, it is seen that the effect on dy­namic factor is very large comparing with the uniform thickness.

The values of the dynamic factor, for a orthotropic composite cylindrical thin shell with thickness vary­ing as a linear function pertaining to h, / ~ = 0.75 , for

different values of the power of time q , of the exter­

nal pressure expression p, for two di fferent approxi-

SOFIYEV e/ at.: BUCKLING OF AN ORTHOTROPIC CYLINDRICAL THIN SHELL 369

mati on function and initial conditions are shown in Table 1. The difference between dynamic factors must be I % in approximation functions. Consequently, for two of the approximation functions nearly the same solutions are obtained. For uniform or non-uniform thickness, when p increases, the value of dynamic factor increases. But when q increases, the value of dynamic factor decreases. For small variations of thickness, it is observed that the difference between dynamic factors of the shells that have uniform and non-uniform thickness is important as shown in Ta­ble 1. For example, when q=l, p=l00 MPa/s and h"h2=0.75, the effect of linear variation of thickness to dynamic factor is 17.5%.

For validity of the analysis, comparisons with iso­tropic and orthotropic cylindrical shells with uniform thickness under external pressure varying linearly de­pendent on time, are made. Comparisons were carried out for the following isotropic material, shell and loading properties: E = 77500 MPa, f.1 = 0.3,

P = 310 kgx S2 Im4, p= 650 MPa/s, q = 1 and ortho­

tropic material, shell and loading properties: EI "E22

=2, 111 2=0.12, p=184 kgxs2/m4, hlR=21143, UR=2.6,

q=l, p=250 MPa/s.

The present theory gives dynamic factor for iso­tropic material Kd=4.248. Experimental value for the same given by Agamirov l3 is Kd=4.0 and numerical result given by Shurnik l4 is Kd=4.239. The present theory gives dynamic factor for orthotropic material Kd=4.32 and numerical result given by Ogibalov et al. 15 is Kd=4.2. It isobserved that the results are har­monic.

Conclusions The buckling of an orthotropic composite thin shell

with continuously varying thickness, subjected to a dynamic loading, has been studied. At first, the fun­damental relations and modified Donnell type dy­namic buckling equations have been written for a or­thotropic composite shell with continuously varying thickness subject to an external pressure which is a power function of time. Then, applying Galerkin's method, a time dependent differential equation with variable coefficient has been obtained. Finally, for different initial conditions and approximation func­tions, applying the Ritz type variational method, ana­lytical expression has been fou nd for the dynamic factor. Using these results, the effects of the variations of the power of time in the external pressure expres-

sion, the ratios of the Young's moduli and the loading parameter on the dynamic factor are studied, numeri­cally, for the case when the thickness of the cylindri ­cal shell varies as a power and exponential function. Furthermore, the present method has been verified by comparisons of dynamic factor with the theoretical and experimental ones given in previous literature for the case of a shell with uniform thickness subject to a uniform external pressure, which is a linear function of time.

Nomenclature

Ell' E22

E o 0 0

el l·e22· eI2

h. flo

i. j

p q

Uij(t)

8 = nrl L

qJ qJij (I)

Ao.AI .A.z J1

J11 2· J121

°1 '(J2 • 03

0"11·0"12·0"22

r p

~I = XII L

~2 = X2 1 r

(.)

Young' s moduli of the orthotropic material

Young' s modulus of the isotropic materi al Axial. circumferential and shear strains on the middle surface of the shell Thickness of the shell and nominal thickness of the shell. respectively Thickness of the shell at the two ends

Wave numbers in the circumferential and axial directions. respectively Power coefficient of thickness variation Dynamic factor

Length of the cylindrical shell

Differential operators defined in Eqs (9-12)

Axial. circumferential and twi sting moments. respectively Axial . circumferential and shear forces. respec­tively Incremental stress resultants about fundamental state Loading parameter Power of time in the external pressure expres­sion Radius of the cylindrical shell Time and critical time. respectively

Displacement of the middle surface in the in­wards normal direction Time dependent amplitude

Coefficient

Stress function and time dependent amplitude. respectively Coefficient defined in Eq. (19)

Poisson's ratio of the isotropic material Poisson's ratios of the orthotropic material

Coefficient defined in Eq. (15)

Stress components

Dimensionless time parameter Density of the material Dimensionless axial coordinate

Dimensionless c ircumferenti al coordinate

Denoting differentiation with respect to

370 INDIAN 1. ENG. MATER. SCI., OCTOBER 2003

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