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Efficient formulation for dynamics of corotational 2D beams
KTH, School of Architecture and the Built Environment (ABE), Civil and Architectural Engineering, Structural Engineering and Bridges.
KTH, School of Architecture and the Built Environment (ABE), Civil and Architectural Engineering, Structural Engineering and Bridges.ORCID iD: 0000-0003-2104-382X
2011 (English)In: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 48, no 2, 153-161 p.Article in journal (Refereed) Published
Abstract [en]

The corotational method is an attractive approach to derive non-linear finite beam elements. In a number of papers, this method was employed to investigate the non-linear dynamic analysis of 2D beams. However, most of the approaches found in the literature adopted either a lumped mass matrix or linear local interpolations to derive the inertia terms (which gives the classical linear and constant Timoshenko mass matrix), although local cubic interpolations were used to derive the elastic force vector and the tangent stiffness matrix. In this paper, a new corotational formulation for dynamic nonlinear analysis is presented. Cubic interpolations are used to derive both the inertia and elastic terms. Numerical examples show that the proposed approach is more efficient than using lumped or Timoshenko mass matrices.

Place, publisher, year, edition, pages
2011. Vol. 48, no 2, 153-161 p.
Keyword [en]
Nonlinear dynamic analysis, Corotational formulation, 2D beam element
National Category
Civil Engineering
Identifiers
URN: urn:nbn:se:kth:diva-37542DOI: 10.1007/s00466-011-0585-6ISI: 000293133800003Scopus ID: 2-s2.0-80052665915OAI: oai:DiVA.org:kth-37542DiVA: diva2:434798
Note
QC 20110816Available from: 2011-08-16 Created: 2011-08-15 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Corotational formulation for nonlinear analysis of flexible beam structures
Open this publication in new window or tab >>Corotational formulation for nonlinear analysis of flexible beam structures
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Flexible beam structures are popular in civil and mechanical engineering. Many of these structures undergo large displacements and finite rotations, but with small deformations. Their dynamic behaviors are usually investigated using finite beam elements. A well known method to derive such beam elements is the corotational approach. This method has been extensively used in nonlinear static analysis. However, its application in nonlinear dynamics is rather limited. The purpose of this thesis is to investigate the nonlinear dynamic behavior of flexible beam structures using the corotational method.

For the 2D case, a new dynamic corotational beam formulation is presented. The idea is to adopt the same corotational kinetic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. This work is presented in the first appended journal paper.

For the 3D case, update procedures of finite rotations, which are central issues in development of nonlinear beam elements in dynamic analysis, are discussed. Three classic and one new formulations of beam elements based on the three different parameterizations of the finite rotations are presented. In these formulations, the corotational method is used to develop expressions of the internal forces and the tangent stiffness matrices, while the dynamic terms are formulated into a total Lagrangian context. Many aspects of the four formulations are investigated. First, theoretical derivations as well as practical implementations are given in details. The similarities and differences between the formulations are pointed out. Second, numerical accuracy and computational efficiency of these four formulations are compared. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. This work is presented in the second appended journal paper.

To make this thesis self-contained, two chapters concerning the parametrization of the finite rotations and the derivation of the 3D corotational beam element in statics are added.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. ix, 35 p.
Series
Trita-BKN. Bulletin, ISSN 1103-4270 ; 115
Keyword
Corotational method, nonlinear dynamic analysis, beam element, large displacements, finite rotations, time stepping method, cubic interpolations
National Category
Engineering and Technology
Identifiers
urn:nbn:se:kth:diva-94880 (URN)
Presentation
2012-05-28, sal B25, Brinellvägen 23, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20120521Available from: 2012-05-16 Created: 2012-05-11 Last updated: 2012-05-21Bibliographically approved
2. Nonlinear dynamics of flexible structures using corotational beam elements
Open this publication in new window or tab >>Nonlinear dynamics of flexible structures using corotational beam elements
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The purpose of this thesis is to develop corotational beam elements for the nonlinear dynamic analyse of flexible beam structures. Whereas corotational beam elements in statics are well documented, the derivation of a corotational dynamic formulation is still an issue.

In the first journal paper, an efficient dynamic corotational beam formulation is proposed for 2D analysis. The idea is to adopt the same corotational kinematic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices.

In the second journal paper, several choices of parametrization and several time stepping methods are compared. To do so, four dynamic formulations are investigated. The corotational method is used to develop expressions of the internal terms, while the dynamic terms are formulated into a total Lagrangian context. Theoretical derivations as well as practical implementations are given in detail. Their numerical accuracy and computational efficiency are then compared. Moreover, four predictors and various possibilities to simplify the tangent inertia matrix are tested.

In the third journal paper, a new consistent beam formulation is developed for 3D analysis. The novelty of the formulation lies in the use of the corotational framework to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix. Cubic interpolations are adopted to formulate both inertia and internal local terms. In the derivation of the dynamic terms, an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained. Four numerical examples are considered to assess the performance of the new formulation against two other ones based on linear interpolations.

Finally, in the fourth journal paper, the previous 3D corotational beam element is extended for the nonlinear dynamics of structures with thin-walled cross-section by introducing the warping deformations and the eccentricity of the shear center. This leads to additional terms in the expressions of the inertia force vector and the tangent dynamic matrix. The element has seven degrees of freedom at each node and cubic shape functions are used to interpolate local transversal displacements and axial rotations. The performance of the formulation is assessed through five examples and comparisons with Abaqus 3D-solid analyses.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. xii, 79 p.
Series
Trita-BKN. Bulletin, ISSN 1103-4270 ; 119
Keyword
corotational method, nonlinear dynamics, large displacements, finite rotations, time stepping method, thin-walled cross-section, beam element
National Category
Construction Management
Identifiers
urn:nbn:se:kth:diva-131701 (URN)
Public defence
2013-10-18, INSA de Rennes, France, 14:00 (English)
Opponent
Supervisors
Note

QC 20131017

Available from: 2013-10-17 Created: 2013-10-17 Last updated: 2013-10-17Bibliographically approved

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