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.

The use of cylindrical shells which have noncircular profiles is common in many fields, such as aerospace, mechanical, civil and marine engineering structures. The displacement buckling modes of thin elastic shells essentially depend on some determining functions such as the radius of the curvature of the neutral surface, the shell thickness, the shape of the shell edges, and so forth. In simple cases when these functions are constant, the buckling modes occupy the entire shell surface. If the determining functions vary from point to point of the neutral surface then localization of the displacement buckling modes lies near the weakest lines on the shell surface, and this kind of problems is too difficult because the radius of its curvature varies with the circumferential coordinate, closed-form or analytic solutions cannot be obtained, in general, for this class of shells, numerical or approximate techniques are necessary for their analysis. Buckling has become more of a problem in recent years since the use of high-strength material requires less material for load support-structures and components have become generally more slender and buckle-prone. Many researchers have considerable interest in the study of stability problems of circular cylindrical shells under uniform axial loads with constant thickness and numerous investigations have been devoted to this, for example [

It has been mentioned in Section

We consider an isotropic, elastic, cylindrical shell of a four-lobed cross section profile expressed by the equation

_{0}_{0}

Coordinate system and geometry of a variable axial loaded cylindrical shell of four-lobed cross section with circumferential variable thickness.

For a general circular cylindrical shell subjected to a non-uniform circumferentially compressive load

The relations between strains and deflections for the cylindrical shells used here are taken from [

For a simply supported shell, the solution of the system of (

The differential equations as shown previously are modified to a suitable form and solved numerically. Hence, by substituting (

The substitution of (

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

Consider the buckling of a four-lobed cross section cylindrical shell with circumferential variable thickness under non-uniform axial loads

The effect of variation in thickness on the buckling loads is presented in Table

The fundamental buckling loads factor

Symmetric Modes | Antisymmetric Modes | |||||||||||||||||

1 | 2 | 5 | 1 | 2 | 5 | |||||||||||||

1 | 40.302 | 94.051 | 2.3 | 50.989 | 136.74 | 2.6 | 88.279 | 250.30 | 2.8 | 94.246 | 112.68 | 1.2 | 168.57 | 284.263 | 1.6 | 388.71 | 953.791 | 2.4 |

2 | 25.574 | 69.646 | 2.7 | 29.288 | 86.071 | 2.9 | 41.987 | 123.78 | 2.9 | 68.699 | 73.642 | 1.1 | 122.94 | 189.365 | 1.5 | 187.28 | 505.348 | 2.7 |

3 | 17.866 | 46.885 | 2.6 | 20.130 | 59.302 | 2.9 | 27.534 | 81.447 | 2.9 | 50.583 | 53.087 | 1.1 | 99.788 | 151.587 | 1.5 | 133.15 | 382.095 | 2.8 |

4 | 15.577 | 2.7 | 17.147 | 50.619 | 2.9 | 22.342 | 66.150 | 2.9 | 43.672 | 46.177 | 1.1 | 76.928 | 1.7 | 101.56 | 294.163 | 2.8 | ||

5 | 43.810 | 2.8 | 65.259 | 138.198 | 2.1 | 84.270 | 244.699 | 2.9 | ||||||||||

6 | 16.712 | 48.118 | 2.9 | 17.841 | 52.776 | 2.9 | 21.554 | 63.860 | 2.9 | 45.836 | 49.448 | 1.1 | 59.274 | 149.957 | 2.5 | 75.065 | 218.302 | 2.9 |

7 | 18.697 | 54.486 | 2.9 | 19.790 | 58.563 | 2.9 | 23.284 | 69.133 | 2.9 | 50.163 | 55.326 | 1.1 | 56.544 | 162.277 | 2.8 | 70.566 | 205.435 | 2.9 |

8 | 21.323 | 62.578 | 2.9 | 22.434 | 66.401 | 2.9 | 25.967 | 76.972 | 2.9 | 51.791 | 63.102 | 1.2 | 162.293 | 2.9 | ||||

9 | 24.502 | 71.840 | 2.9 | 25.667 | 75.978 | 2.9 | 29.305 | 86.887 | 2.9 | 52.878 | 72.511 | 1.3 | 56.837 | 165.161 | 2.9 | 69.586 | 202.863 | 2.9 |

10 | 28.184 | 82.184 | 2.9 | 29.431 | 87.127 | 2.9 | 33.251 | 98.609 | 2.9 | 54.918 | 83.406 | 1.5 | 58.897 | 171.260 | 2.9 | 71.655 | 208.993 | 2.9 |

The results presented in this table show that the increase of the thickness ratio tends to increase the critical buckling load (

When a structure subjected usually to compression undergoes visibly large displacement transverse to the load then it is said to buckle, and for small loads the buckle is elastic since buckling displacements disappear when the loads are removed. Generally, the buckling displacements mode is located at the weakest generatrix of the shell where the unsteady axial compression is a maximum, and the shell has less stiffness. Figures

The symmetric buckling deformations of a cylindrical shell of a four-lobed cross section with variable thickness.

The antisymmetric buckling deformations of a cylindrical shell of a four-lobed cross section with variable thickness.

We consider a special case for a circular cylindrical shell (

The fundamental buckling loads factor

Symmetric & Antisymmetric Modes | |||||||||

1 | 2 | 5 | |||||||

1 | 143.814 | 189.854 | 457.252 | 2.4 | 335.448 | 894.988 | 2.6 | ||

2 | 135.864 | 295.683 | 2.1 | 164.134 | 432.286 | 2.6 | 241.186 | 673.852 | 2.7 |

3 | 130.580 | 301.418 | 2.3 | 151.581 | 411.782 | 2.7 | 204.525 | 581.959 | 2.8 |

4 | 127.165 | 404.754 | 3.1 | 143.553 | 396.756 | 2.7 | 183.601 | 527.618 | 2.8 |

5 | 124.319 | 309.743 | 2.5 | 137.783 | 385.323 | 2.7 | 169.798 | 491.308 | 2.8 |

6 | 122.243 | 312.076 | 2.5 | 133.457 | 376.613 | 2.8 | 159.812 | 464.874 | 2.9 |

7 | 120.566 | 315.157 | 2.6 | 130.102 | 369.922 | 2.8 | 152.083 | 444.416 | 2.9 |

8 | 119.173 | 314.672 | 2.6 | 127.390 | 364.647 | 2.8 | 145.726 | 427.695 | 2.9 |

9 | 117.985 | 312.156 | 2.6 | 125.647 | 360.347 | 2.8 | 140.133 | 413.152 | 2.9 |

10 | 116.947 | 293.795 | 2.5 | 123.156 | 356.685 | 2.8 | 134.853 | 399.486 | 2.9 |

Figure

The circumferential buckling modes of a circular cylindrical shell with variable thickness.

Figure

Critical buckling loads versus thickness ratio of a four-lobed cross section cylindrical shell with variable thickness, (

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

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