#### Abstract

The static buckling of a cylindrical shell of a four-lobed cross section of variable thickness subjected to non-uniform circumferentially compressive loads is investigated based on the thin-shell theory. Modal displacements of the shell can be described by trigonometric functions, and Fourier's approach is used to separate the variables. The governing equations of the shell are reduced to eight first-order differential equations with variable coefficients in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the nonlinear differential equations of the cylindrical shells by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The transfer matrix approach is used to get the critical buckling loads and the buckling deformations for symmetrical and antisymmetrical shells. Computed results indicate the sensitivity of the critical loads and corresponding buckling modes to the thickness variation of cross section and the radius variation at lobed corners of the shell.

#### 2. Theory and Formulation of the Problem

It has been mentioned in Section 1 that the problem structure is modeled by thin-shell theory. In order to have a better representation, the shell geometry and governing equations are modeled as separate parts. The formulation of these parts is presented below.

##### 2.1. Geometrical Formulation

We consider an isotropic, elastic, cylindrical shell of a four-lobed cross section profile expressed by the equation , whereis the varied radius along the cross section midline, is the reference radius of curvature, chosen to be the radius of a circle having the same circumference as the four-lobed profile, and is a prescribed function of and can be described as

and are the axial and circumferential lengths of the middle surface of the shell, and the thickness is varying continuously in the circumferential direction. The cylindrical coordinates are taken to define the position of a point on the middle surface of the shell, as shown in Figure 1(a), and Figure 1(b) shows the four-lobed cross section profile of the middle surface, with the apothem denoted by and the radius of curvature at the lobed corners by . While and are the deflection displacements of the middle surface of the shell in the longitudinal, circumferential and transverse directions, respectively. We suppose that the shell thickness at any point along the circumference is small and depends on the coordinate and takes the following form: where h0 is a small parameter, chosen to be the average thickness of the shell over the length . For the cylindrical shell which its cross section is obtained by the cutaway the circle of the radius from the circle of the radius R0 (see Figure 1(c)) function has the form: whereis the amplitude of thickness variation, = / , and is the distance between the circles centers. In general case is the minimum value of while is the maximum value of , and in case of = 0 the shell has constant thickness . The dependence of the shell thickness ratio = / on has the form.

##### 2.2. Governing Equations

For a general circular cylindrical shell subjected to a non-uniform circumferentially compressive load , the static equilibrium equations of forces, based on the Goldenveizer-Novozhilov theory [26, 27], can be shown to be of the following forms:

where and are the normal and transverse shearing forces in the and directions, respectively, and are the in-plane shearing forces, and are the bending moment and the twisting moment, respectively, is the equivalent (Kelvin-Kirchoff) shearing force, R is the radius of curvature of the middle surface, and . We assume that the shell is loaded along the circumferential coordinate with non-uniform axial loads which vary with, where the compressive load does not reach its critical value at which the shell loses stability. Generally, the form of the non-uniform load may be expressed as:

where is a given function of and is a constant. We assume that the shell is loaded by axially non-uniform loadsand takes the form as in : and the sketch depicting this load is given in Figure 1(d). The applied load in this form establishes two zones on the shell surface: one is the compressive zone, , for () where the buckling load factor is a maximum and the thickness is a minimum and the other is the tensile zone, , for () where the buckling load factor is a minimum and the thickness is a maximum, as shown in this figure. Note that in the case of applied uniform axial loads. Hereby, we deduce the following ratio of critical loads:

is the lowest value of applied compressive loads and named by the critical load.

The relations between strains and deflections for the cylindrical shells used here are taken from  as follows:

where and are the normal strains of the middle surface of the shell, and are the shear strains, and the quantities and representing the change of curvature and the twist of the middle surface, is the bending slope, and is the angular rotation. The components of force and moment resultants in terms of (2.8) are given as: From (2.4)–(2.9), with eliminating the variables and which are not differentiated with respect to , the system of the partial differential equations for the state variables and of the shell is obtained as follows:

The quantitiesand, respectively, are the extensional and flexural rigidities expressed in terms of the Young’s modulus E, Poisson’s ratio and the wall thickness as the form: and , and on considering the variable thickness of the shell, using (2.3), they take the form:

where and are the reference extensional and flexural rigidities of the shell, chosen to be the averages on the middle surface of the shell over the length .

For a simply supported shell, the solution of the system of (2.10) is sought as follows:

where is the axial half-wave number, and the quantities are the state variables and undetermined functions of.

#### 3. Matrix Form of the Governing Equations

The differential equations as shown previously are modified to a suitable form and solved numerically. Hence, by substituting (2.12) into (2.10), after appropriate algebraic operations and taking relations (2.11) into account, the system of buckling equations of the shell can be written in nonlinear ordinary differential equations referred to the variable only are obtained, in the following matrix form:

By using the state vector of fundamental unknowns, system (3.1) can be written as:

For the noncircular cylindrical shell which cross section profile is obtained by function (), the hypotenuse () of a right triangle whose sides are infinitesimal distances along the surface coordinates of the shell takes the following form: then we have

Using (3.4), the system of buckling equations (3.2) takes the following form:

where , and the coefficients matrix are given as:

in terms of the following dimensionless shell parameters: curvature parameter , buckling load factor, , and . The state vector of fundamental unknowns can be easily expressed as: by using the transfer matrix of the shell, the substitution of the expression into (3.5) yields:

The governing system of buckling (3.8) is too complicated to obtain any closed-form solution, and this problem is highly favorable for solving by numerical methods. Hence, the matrix is obtained by using numerical integration, by use of the Runge-kutta integration method of forth-order, with the starting value (unit matrix) which is given by taking in (3.7), and its solution depends only on the geometric and martial properties of the shell. For a plane passing through the central axis in a shell with structural symmetry, symmetrical and antisymmetrical profiles can be obtained, and consequently, only one-half of the shell circumference is considered with the boundary conditions at the ends taken to be the symmetric or antisymmetric type of buckling deformations. Therefore, the boundary conditions for symmetrical and antisymmetrical bucking deformations are

#### 4. Buckling Loads and Buckling Modes

The substitution of (3.9) into (3.7) results in the following buckling equations:

The matrices [] depend on the buckling load factorand the circumferential angle θ. Equation (4.1) gives a set of linear homogenous equations with unknown coefficients and, respectively, at . For the existence of a nontrivial solution of these coefficients, the determinant of the coefficient matrix should be vanished. The standard procedures cannot be employed for obtaining the eigenvalues of the load factor. The nontrivial solution is found by searching the valueswhich make the determinant zero by using Lagrange interpolation procedure. The critical buckling load of the shell will be the smallest member of this set of values. The buckling deformations (circumferential buckling displacement mode) at any point of the cross section of the shell, for each axial half mode m, are determined by calculating the eigenvectors corresponding to the eigenvalues by using Gaussian elimination procedure.

#### 5. Computed Results and Discussion

A computer program based on the analysis described herein has been developed to study the buckling behaviour of the shell under consideration. The critical buckling loads and the corresponding buckling deformations of the shell are calculated numerically, and some of the results shown next are for cases that have not as yet been considered in the literature. Our study is divided into two parts in which the Poisson’s ratio takes the value 0.3.

##### 5.1. Buckling Results

Consider the buckling of a four-lobed cross section cylindrical shell with circumferential variable thickness under non-uniform axial loads, varying over the length. The study of shell buckling is determined by finding the load factor which equals the eigenvalues of (4.1) for each value of m, separately. To obtain the buckling loads = we will search the set of all eigenvalues, and to obtain the critical buckling loads , which correspond to loss of stability of the shell, we will search the lowest values of this set. The numerical results presented herein pertain to the buckling loads in the case of uniform and non-uniform loads for symmetric and antisymmetric type-modes.

The effect of variation in thickness on the buckling loads is presented in Table 1 which gives the fundamental buckling loads factor of a four-lobed cross section cylindrical shell with radius ratio= 0.5 versus the axial half-wave number m for the specific values of thickness ratio symmetric and antisymmetric type-modes. A-columns and B-columns correspond to applied non-uniform and uniform axial loads, respectively.

##### 5.3. Particular Case

Figure 4 shows the circumferential buckling modes of a circular cylindrical shell with variable thickness under the specific load. It is seen from this figure that the buckling deformations for applied uniform loads are distributed regularly over the shell surface of constant thickness, see (i), (ii) in Figure 4. These figures are in quite good agreement with . It can also be seen from this figure that the shell of applied non-uniform loads buckles more easily than one of applied uniform loads.

Figure 5 shows the variations in the critical buckling loads of a non-uniformly loaded shell of a four-lobed cross section versus the radius ration , for the specific values of thickness ratio η. The axial half-wave number of corresponding critical buckling loads is shown in this figure as (m). It is seen from this figure, for the symmetric and antisymmetric type-modes, that an increase in the radius ratio causes an increase in the critical loads, and when the foregoing ratio becomes unity the latter quantities take the same values and are assumed to be for a circular cylindrical shell. It is observed that the critical loads increase with an increase in the thickness ratio where the shell becomes more stiffness. Upon increasing the radius ratio, the critical buckling axial half-wave number increases. The nominal axial half-wave number corresponding to the critical buckling load may be in general depends on the radius of curvature at the lobed corners of the shell.

#### 6. Conclusions

An approximate analysis for studying the elastic buckling characteristics of circumferentially non-uniformly axially loaded cylindrical shell of a four-lobed cross section having circumferential varying thickness is presented. The computed results presented herein pertain to the buckling loads and the corresponding mode shapes of buckling displacements by using the transfer matrix approach. The method is based on thin-shell theory and applied to a shell of symmetric and antisymmetric type-modes, and the analytic solutions are formulated to overcome the mathematical difficulties associated with mode coupling caused by variable shell wall curvature and thickness. The fundamental buckling loads and corresponding buckling deformations have been presented, and the effects of the thickness ratio of the cross-section and the non-uniformity of applied load on the critical loads and buckling modes were examined.

The study showed that the buckling strength for non-uniform loads was lower than that under uniform axial loads. The deformation of corresponding buckling load are located at the compressive zone of a small thickness but, in contrast, the deformation of corresponding critical load are located at the tensile zone of a large thickness, and this indicates the possibility of a static loss of stability for the shell at values of less than the critical value. Generally, the symmetric and antisymmetric buckling deformations take place in the less stiffened zones of the shell surface where the lobes are located. However, for the applied uniform and non-uniform axial loads, the critical buckling loads increase with either increasing radius ratio or increasing thickness ratio and become larger for a circular cylindrical shell.

#### Acknowledgment

The author is grateful to anonymous reviewers for their good efforts and valuable comments which helped to improve the quality of this paper.