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Nonlinear dynamics of flexible structures using corotational beam elements
KTH, School of Architecture and the Built Environment (ABE), Civil and Architectural Engineering, Structural Engineering and Bridges.
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 [en]
corotational method, nonlinear dynamics, large displacements, finite rotations, time stepping method, thin-walled cross-section, beam element
National Category
Construction Management
Identifiers
URN: urn:nbn:se:kth:diva-131701OAI: oai:DiVA.org:kth-131701DiVA: diva2:656768
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
List of papers
1. Efficient formulation for dynamics of corotational 2D beams
Open this publication in new window or tab >>Efficient formulation for dynamics of corotational 2D beams
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.

Keyword
Nonlinear dynamic analysis, Corotational formulation, 2D beam element
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-37542 (URN)10.1007/s00466-011-0585-6 (DOI)000293133800003 ()2-s2.0-80052665915 (Scopus ID)
Note
QC 20110816Available from: 2011-08-16 Created: 2011-08-15 Last updated: 2017-12-08Bibliographically approved
2. Dynamics of 3D beam elements in a corotational context: A comparative study of established and new formulations
Open this publication in new window or tab >>Dynamics of 3D beam elements in a corotational context: A comparative study of established and new formulations
2012 (English)In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 61, 97-111 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with Newmark time stepping methods and finite rotations for nonlinear finite element analysis of flexible beam structures. The corotational method is used to develop expressions of the internal forces and the corresponding tangent stiffness matrices. For the dynamic part, four formulations based on different parameterizations of rotations are compared. The first three are classic formulations taken from the literature with some modifications for two of them. The last one is new and uses three of the four Euler parameters (quaternion) as rotational variables. For all these approaches, theoretical derivations as well as practical implementations are given in detail. The similarities and differences between the formulations are pointed out. Six numerical examples are studied in order to compare these four formulations in terms of numerical accuracy and computational efficiency. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. The numerical results show that these four formulations have the same numerical accuracy, but that the computational efficiency depends on the choice of the tangent inertia matrix. Besides, the new formulation proposed in this paper turns out to be the fastest one.

Keyword
Time stepping method, finite rotations, nonlinear dynamics, 3D beam elements
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-95197 (URN)10.1016/j.finel.2012.06.007 (DOI)000307825900010 ()2-s2.0-84864428298 (Scopus ID)
Note

QC 20120829. Updated from submitted to published.

Available from: 2012-05-16 Created: 2012-05-16 Last updated: 2017-12-07Bibliographically approved
3. A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures
Open this publication in new window or tab >>A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures
2014 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 269, 538-565 p.Article in journal (Refereed) Published
Abstract [en]

The purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3D flexible frames. The novelty of the formulation lies in the use of the corotational framework (i.e., the decomposition into rigid body motion and pure deformation) to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix. As a consequence, 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. To enhance the efficiency of the iterative procedure, an approximate expression of the tangent dynamic matrix is adopted. Four numerical examples are considered to assess the performance of the new formulation against the one suggested by Simo and Vu-Quoc (1988) [48]. It was observed that the proposed formulation proves to combine accuracy with efficiency. In particular, the present approach achieves the same level of accuracy as the formulation of Simo and Vu-Quoc but with a significantly smaller number of elements.

Keyword
3D beam elements, Corotational method, Finite rotations, Nonlinear dynamics
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-131698 (URN)10.1016/j.cma.2013.11.007 (DOI)000330815900024 ()2-s2.0-84890860488 (Scopus ID)
Note

QC 20140204. Updated from manuscript to article in journal.

Available from: 2013-10-17 Created: 2013-10-17 Last updated: 2017-12-06Bibliographically approved
4. Corotational formulation for nonlineardynamics of beams with arbitrary thin-walled open cross-sections
Open this publication in new window or tab >>Corotational formulation for nonlineardynamics of beams with arbitrary thin-walled open cross-sections
(English)Manuscript (preprint) (Other academic)
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-131699 (URN)
Note

QS 2013

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

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