^{1}

^{2}

^{*}

^{3}

^{4}

In present work, post-buckling behavior of imperfect (of eigen form) laminated composite cylindrical shells with different L/D and R/t ratios subjected to axial, bending and torsion loads has been investigated by using an equilibrium path approach in the finite element analysis. The Newton-Raphson approach as well as the arc-length approach is used to ensure the correctness of the equilibrium paths up to the limit point load. Post-buckling behavior of imperfect cylindrical shells with different L/D and R/t ratios of interest is obtained and the theoretical knock-down factors are reported for the considered cylindrical shells.

Cylindrical shells are very often used as primary load carrying structural members in aerospace, civil, mechanical and nuclear engineering fields and these structural elements are very often subjected to multiple states of loading simultaneously such as the combination of axial, bending and torsion loads. The mechanical behavior of the above structural elements is extremely sensitive to the presence of geometric imperfections and the modeling of these geometric imperfections plays a pivotal role in accurately understanding the mechanical behavior by means of either an analytical or finite element analysis approaches. The complex behavior of these imperfect composite cylindrical shells subjected to axial compressive [

Recently authors have carried out an extensive literature review in References [

Karyadi [

Yamaki [

The exhaustive study on post-buckling behavior of laminated composite cylindrical shells subjected to general fundamental load cases such as axial, bending and torsion loads with different L/D and R/t ratios is meagerly seen in the available open literature. The proposed piece of work makes a modest attempt to bridge this gap with an endeavor to understand the post-buckling behavior of the laminated composite cylindrical shells subjected to these fundamental loads.

The finite element discretization process for geometrically non-linear analysis yields a set of simultaneous equations:

where

where

When buckling occurs, the external loads do not change, i.e.,

where

Nonlinear analysis of a geometrically perfect or imperfect cylindrical shell using Newton-Raphson approach generally involves the determination of the equilibrium path up to the limit point load and beyond which the slope of the load-deflection curve (or equilibrium path) ceases to be positive. Post-buckling analysis by means of an arc-length approach generally involves the determination of the full equilibrium path which also includes tracing of the unstable solution of the equilibrium path. Salient steps involved in arriving at the post-buckling behavior of imperfect cylindrical shells are briefly summarized below:

Steps Followed in Post-Buckling Analysis1) The linear buckled mode shape has been chosen as the basis of initial imperfection. Magnitude of initial

imperfection is referred with reference to the thickness (t) parameter of the cylindrical shell. It must be noted that the shape of imperfection can be given in the form of linear combination of buckled mode shapes or random imperfection or experimentally measured imperfection shape.

2) After applying initial geometric imperfection, a nonlinear analysis has been performed to trace the equilibrium path of interest.

3) Nonlinear analysis involves the application of either Newton-Raphson approach or an arc-length approach to solve Equation (1).

4) Load-deflection curve obtained from the Newton-Raphson approach represents the primary equilibrium path where as the load-deflection curve traced in an arc-length approach includes the primary (stable) as well as secondary (unstable) equilibrium paths. All the proposed results use an arc-length method for post-buckling analysis results unless otherwise it is explicitly mentioned.

In present work, post-buckling behavior of the laminated composite cylindrical shells subjected to axial, bending and torsion loads has been investigated by using the post-buckling analysis.

Throughout this study, the cylindrical shell is assumed to be laminated composite cylindrical shell made up of E-glass/polyester resin and the material properties are directly taken from Ref. [

Material Properties | Values |
---|---|

E_{11} | 149.6 GPa |

G_{23} | 2.5 GPa |

ν_{23} | 0.45 |

E_{22} = E_{33} | 9.9 GPa |

G_{12} = G_{13} | 4.5 GPa |

ν_{12} = ν_{13} | 0.28 |

cal results obtained from the Newton Raphson approach as well as the arc-length approach for a typical imperfection magnitude (ξ = w*/t, where w* is the maximum imperfection amplitude and t is the thickness of the cylindrical shell) of laminated composite cylindrical shell subjected to axial compressive and bending load respectively. In general, it is observed that these two approaches have shown good agreement in predicting the primary equilibrium path as well as in predicting the limit point load of the considered composite cylindrical shells which in turn poses as a benchmark validation for all the results discussed in this paper. For the sake of better clarity to the reader, only post-buckling analysis results obtained from the arc-length approach are only discussed subsequently in the paper.

L/D | R/t | Z | P_{cr} (in KN) | P/P_{cr} | ||
---|---|---|---|---|---|---|

ξ = 0.1 | ξ = 0.5 | ξ = 1.0 | ||||

5 | 500 | 47,650 | 7.077 | 0.918 | 0.798 | 0.660 |

2.5 | 500 | 11,913 | 7.214 | 0.854 | 0.578 | 0.502 |

1 | 500 | 1906 | 7.350 | 0.773 | 0.535 | 0.421 |

0.5 | 500 | 477 | 7.612 | 0.768 | 0.450 | 0.334 |

5 | 100 | 9530 | 157.803 | 0.824 | 0.890 | 0.812 |

2.5 | 100 | 2383 | 174.247 | 0.781 | 0.632 | 0.637 |

1 | 100 | 381 | 177.716 | 0.776 | 0.530 | 0.485 |

0.5 | 100 | 95 | 196.158 | 0.750 | 0.521 | 0.442 |

eigen imperfection magnitudes and in

Numerical results (Linear or Critical torsional buckling loads) obtained from the present work are first validated in

L/D | R/t | Z | M_{cr} (N-m) | M _{/}M_{cr} | ||
---|---|---|---|---|---|---|

ξ = 0.1 | ξ = 0.5 | ξ = 1.0 | ||||

5 | 500 | 47,650 | 566.47 | 0.621 | 0.407 | 0.347 |

2.5 | 500 | 11,913 | 566.77 | 0.669 | 0.412 | 0.355 |

1 | 500 | 1906 | 577.12 | 0.731 | 0.444 | 0.364 |

0.5 | 500 | 477 | 594.95 | 0.762 | 0.495 | 0.395 |

5 | 100 | 9530 | 13548 | 0.548 | 0.432 | 0.379 |

2.5 | 100 | 2383 | 13633 | 0.638 | 0.441 | 0.390 |

1 | 100 | 381 | 14159 | 0.738 | 0.491 | 0.393 |

0.5 | 100 | 95 | 15336 | 0.758 | 0.559 | 0.457 |

Linear Buckling Load, psi (or Mpa) | |||
---|---|---|---|

Mesh | Present | Park et al. [ | Donnell |

10 × 21 | 10,940 (75.428 Mpa) | 10,903 (75.17 Mpa) | 10559 (72.80 Mpa) |

15 × 30 | 10,767 (74.23 Mpa) | 10,735 (74.02 Mpa) | - |

L/D | Linear Torsional Buckling Load Present (N/m) | Linear Torsional Buckling Load Ref. [ |
---|---|---|

L/D = 0.5 | 0.1718 × 10^{6} | 0.1576 × 10^{6} |

L/D = 2.5 | 0.0816 × 10^{6} | 0.0757 × 10^{6} |

drical shells.

In all

L/D | Torsional Buckling Load (Present) (N/m) | Torsional Buckling Load Ref. [ |
---|---|---|

L/D = 0.5 | 0.3281 × 10^{6} | 0.3030 × 10^{6} |

L/D = 2.5 | 0.0681 × 10^{6} | 0.0613 × 10^{6} |

Material Properties | Values |
---|---|

E_{11} | 149.6 GPa |

G_{23} | 2.5 GPa |

ν_{23} | 0.45 |

E_{22} = E_{33} | 9.9 GPa |

G_{12} = G_{13} | 4.5 GPa |

ν_{12} = ν_{13} | 0.28 |

L/D | R/t | Z | T_{cr} (N-m) | T/T_{cr} | ||
---|---|---|---|---|---|---|

ξ = 0.1 | ξ = 0.5 | ξ = 1.0 | ||||

2.5 | 500 | 11913 | 1646.2 | 0.975 | 0.904 | 0.849 |

0.5 | 500 | 477 | 5465.5 | 0.909 | 0.754 | 0.767 |

2.5 | 100 | 2383 | 262320 | 0.948 | 0.845 | 0.754 |

0.5 | 100 | 95 | 71688 | 0.863 | 0.697 | 0.645 |

Type of Load | L/D | R/t | Z | P_{cr}/M_{cr}/T_{cr} | P/P_{cr}/M/M_{cr}/T/T_{cr} | ||
---|---|---|---|---|---|---|---|

ξ = 0.1 | ξ = 0.5 | ξ = 1.0 | |||||

Axial compressive load | 5.0 | 500 | 47650 | 7077 N | 0.918 | 0.798 | 0.660 |

0.5 | 500 | 477 | 7612 N | 0.768 | 0.450 | 0.334 | |

Bending load | 5.0 | 500 | 47650 | 566.5 N-m | 0.621 | 0.407 | 0.347 |

0.5 | 500 | 477 | 594.9 N-m | 0.762 | 0.495 | 0.395 | |

Torsion load | 5.0 | 500 | 47650 | 1050.0 N-m | 0.974 | 0.867 | 0.819 |

0.5 | 500 | 477 | 5465.5 N-m | 0.909 | 0.754 | 0.767 |

Post-buckling behavior of laminated imperfect (of eigen form) composite cylindrical shells subjected to different fundamental loads is investigated by using an equilibrium path approach. Numerical results obtained from the proposed study are validated with the available literature values wherever they are found applicable, apart from providing comprehensive post-buckling analysis results subjected to different fundamental loads. The sensitivity of the limit point loads (or knock-down factors) for various cylindrical shells with different fundamental loading conditions is clearly discussed.

Proposed study finds an immediate application in fundamental understanding of the mode jumping phenomenon which is commonly associated with the behavior of cylindrical shells subjected to different combination of loads as well as material properties. The confidence gained on this study can also be extended to investigate the influence of either random imperfection shapes (or) actually measured imperfection shapes. The detailed outcome of the above studies will be reported in future works planned by the authors’.

Yengula VenkataNarayana,Jagadish BabuGunda,Ravinder ReddyPinninti,MarkandeyaRavvala, (2015) Post-Buckling Behavior of Laminated Composite Cylindrical Shells Subjected to Axial, Bending and Torsion Loads. World Journal of Engineering and Technology,03,185-194. doi: 10.4236/wjet.2015.34019