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Corotational formulation for nonlinear analysis of flexible beam structures
KTH, School of Architecture and the Built Environment (ABE), Civil and Architectural Engineering, Structural Engineering and Bridges.
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 [en]
Corotational method, nonlinear dynamic analysis, beam element, large displacements, finite rotations, time stepping method, cubic interpolations
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:kth:diva-94880OAI: oai:DiVA.org:kth-94880DiVA: diva2:526302
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
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

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