### Abstract

Generalised Beam Theory (GBT), intended to analyse the structural behaviour of prismatic thin-walled

members and structural systems, expresses the member local and/or global deformed configuration as a

combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude

functions. The determination of the latter, usually the most computer-intensive step of the analysis, is almost

always performed by means of GBT-based conventional 1D (beam) finite elements, using Hermite cubic

polynomials as shape functions. This paper presents the formulation, implementation and application of

a new GBT-based exact element, developed in the context of member (linear) buckling analyses. This exact

element, originally proposed by Eisenberger (1990), approximates the modal longitudinal amplitude

functions by means of power series, whose coefficients are obtained by means of a recursive formula –

since the higher-order coefficients tend to vanish, the method has the potential to become exact (up to computer precision). The buckling load parameters are the solutions of the (highly) non-linear characteristic equation associated with the buckling eigenvalue problem. A few numerical illustrative examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of buckling solutions with the exact and standard GBT-based (finite) elements. This comparison provides evidence that the exact element leads to equally accurate results with less degrees of freedom and, moreover, without the need to define a (longitudinal) mesh the relative efficiency of the exact element is higher when the buckling modes exhibit larger half-wave numbers.

members and structural systems, expresses the member local and/or global deformed configuration as a

combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude

functions. The determination of the latter, usually the most computer-intensive step of the analysis, is almost

always performed by means of GBT-based conventional 1D (beam) finite elements, using Hermite cubic

polynomials as shape functions. This paper presents the formulation, implementation and application of

a new GBT-based exact element, developed in the context of member (linear) buckling analyses. This exact

element, originally proposed by Eisenberger (1990), approximates the modal longitudinal amplitude

functions by means of power series, whose coefficients are obtained by means of a recursive formula –

since the higher-order coefficients tend to vanish, the method has the potential to become exact (up to computer precision). The buckling load parameters are the solutions of the (highly) non-linear characteristic equation associated with the buckling eigenvalue problem. A few numerical illustrative examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of buckling solutions with the exact and standard GBT-based (finite) elements. This comparison provides evidence that the exact element leads to equally accurate results with less degrees of freedom and, moreover, without the need to define a (longitudinal) mesh the relative efficiency of the exact element is higher when the buckling modes exhibit larger half-wave numbers.

Original language | English |
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Title of host publication | Proceedings of the Coupled Instabilities in Metal Structures Conference |

Pages | 1-24 |

Number of pages | 24 |

DOIs | |

Publication status | Published - 2016 |

Event | Coupled Instabilities in Metal Structures Conference 2016 - Baltimore, United States Duration: 7 Nov 2016 → 8 Nov 2016 |

### Conference

Conference | Coupled Instabilities in Metal Structures Conference 2016 |
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Country | United States |

City | Baltimore |

Period | 7/11/16 → 8/11/16 |

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## Cite this

Bebiano, R., Eisenberger, M., Camotim, D., & Gonçalves, R. D. M. (2016). GBT-based buckling analysis using the exact element method. In

*Proceedings of the Coupled Instabilities in Metal Structures Conference*(pp. 1-24). [21] https://doi.org/10.1142/S0219455417501255