CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Energy-momentum conserving time-stepping algorithms for nonlinear dynamics of planar and spatial Euler-Bernoulli/Timoshenko beams
KTH, School of Architecture and the Built Environment (ABE), Civil and Architectural Engineering, Structural Engineering and Bridges. INSA de Rennes.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Large deformations of flexible beams can be described using either the co-rotational approach or the total Lagrangian formalism. The co-rotational method is an attractive approach to derive highly nonlinear beam elements because it combines accuracy with numerical efficiency. On the other hand, the total Lagrangian formalism is the natural setting for the construction of geometrically exact beam theories. Classical time integration methods such as Newmark, standard midpoint rule or the trapezoidal rule do suffer severe shortcomings in nonlinear regimes. The construction of time integration schemes for highly nonlinear problems which conserve the total energy, the momentum and the angular momentum is addressed for planar co-rotational beams and for a geometrically exact spatial Euler-Bernoulli beam.

In the first part of the thesis, energy-momentum conserving algorithms are designed for planar co-rotational beams. Both Euler-Bernoulli and Timoshenko kinematics are addressed. These formulations provide us with highly complex non-linear expressions for the internal energy as well as for the kinetic energy which involve second derivatives of the displacement field. The main idea of the algorithm is to circumvent the complexities of the geometric non-linearities by resorting to strain velocities to provide, by means of integration, the expressions for the strain measures themselves. Similarly, the same strategy is applied to the highly nonlinear inertia terms. Several examples have been considered in which it was observed that energy, linear momentum and angular momentum are conserved for both formulations even when considering very large number of time-steps. Next, 2D elasto-(visco)-plastic fiber co-rotational beams element and a planar co-rotational beam with generalized elasto-(visco)-plastic hinges at beam ends have been developed and compared against each other for impact problems. Numerical examples show that strain rate effects influence substantially the structure response.

In the second part of this thesis, a geometrically exact 3D Euler-Bernoulli beam theory is developed. The main challenge in defining a three-dimensional Euler-Bernoulli beam theory lies in the fact that there is no natural way of defining a base system at the deformed configuration. A novel methodology to do so leading to the development of a spatial rod formulation which incorporates the Euler-Bernoulli assumption is provided. The approach makes use of Gram-Schmidt orthogonalisation process coupled to a one-parametric rotation to complete the description of the torsional cross sectional rotation and overcomes the non-uniqueness of the Gram-Schmidt procedure. Furthermore, the formulation is extended to the dynamical case and a stable, energy conserving time-stepping algorithm is developed as well. Many examples confirm the power of the formulation and the integration method presented.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2018. , p. 67
Series
TRITA-ABE-DLT ; 1836
Keywords [en]
Nonlinear Dynamics, Energy-momentum conserving scheme, 2D co-rotational beam, Geometrically exact 3D Euler-Bernoulli beam, impact
National Category
Building Technologies
Research subject
Civil and Architectural Engineering
Identifiers
URN: urn:nbn:se:kth:diva-238598ISBN: 978-91-7873-023-0 (print)OAI: oai:DiVA.org:kth-238598DiVA, id: diva2:1261058
Public defence
2018-12-11, Amphi GCU, INSA de Rennes, 20 avenue des Buttes de Coësmes CS 70839 City: Rennes, France, Rennes, 15:00 (English)
Opponent
Supervisors
Note

QC 20181106

Available from: 2018-11-06 Created: 2018-11-06 Last updated: 2018-11-06Bibliographically approved
List of papers
1. An energy-momentum co-rotational formulation for nonlinear dynamics of planar beams
Open this publication in new window or tab >>An energy-momentum co-rotational formulation for nonlinear dynamics of planar beams
2017 (English)In: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 187, p. 50-63Article in journal (Refereed) Published
Abstract [en]

This article presents an energy-momentum integration scheme for the nonlinear dynamic analysis of planar Euler-Bernoulli beams. The co-rotational approach is adopted to describe the kinematics of the beam and Hermitian functions are used to interpolate the local transverse displacements. In this paper, the same kinematic description is used to derive both the elastic and the inertia terms. The classical midpoint rule is used to integrate the dynamic equations. The central idea, to ensure energy and momenta conservation, is to apply the classical midpoint rule to both the kinematic and the strain quantities. This idea, developed by one of the authors in previous work, is applied here in the context of the co-rotational formulation to the first time. By doing so, we circumvent the nonlinear geometric equations relating the displacement to the strain which is the origin of many numerical difficulties. It is rigorously shown that the proposed method conserves the total energy of the system and, in absence of external loads, the linear and angular momenta remain constant. The accuracy and stability of the proposed algorithm, especially in long term dynamics with a very large number of time steps, is assessed through four numerical examples.

Place, publisher, year, edition, pages
Elsevier Ltd, 2017
Keywords
2D beam, Co-rotational formulation, Conserving energy, Energy-momentum method, Nonlinear dynamic, Dynamics, Kinematics, Momentum, Energy momentum method, Euler Bernoulli beams, Geometric equations, Long term dynamics, Transverse displacements, Nonlinear equations
National Category
Civil Engineering
Identifiers
urn:nbn:se:kth:diva-207289 (URN)10.1016/j.compstruc.2017.03.021 (DOI)000401675600004 ()2-s2.0-85017346411 (Scopus ID)
Note

QC 20170619

Available from: 2017-06-19 Created: 2017-06-19 Last updated: 2018-12-05Bibliographically approved
2. Energy-momentum method for co-rotational plane beams: A comparative study of shear flexible formulations
Open this publication in new window or tab >>Energy-momentum method for co-rotational plane beams: A comparative study of shear flexible formulations
2017 (English)In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 134, p. 41-54Article in journal (Refereed) Published
Abstract [en]

This paper presents an energy-momentum method for three dynamic co-rotational formulations of shear flexible 2D beams. The classical midpoint rule is applied for both kinematic and strain quantities. Although the idea as such was developed in previous work, its realization and testing in the context of co-rotational Timoshenko 2D beam elements is done here for the first time. The main interest of the method is that the total energy and momenta are conserved. The three proposed formulations are based on the same co-rotational framework but they differ in the assumptions done to derive the local formulations. Four numerical applications are used to assess the accuracy and efficiency of each formulation. In particularly, the conservation of energy with a very large number of steps and the possibility to simplify the tangent dynamic matrix are investigated.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
2D beams, Co-rotational formulation, Energy-momentum method, Nonlinear dynamics, Shear
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-210189 (URN)10.1016/j.finel.2017.04.001 (DOI)000405655800004 ()2-s2.0-85020725160 (Scopus ID)
Note

QC 20170629

Available from: 2017-06-29 Created: 2017-06-29 Last updated: 2018-11-06Bibliographically approved
3. Co-rotating flexible beam with generalized visco-plastic hinges for the nonlinear dynamics of frame structures under impacts
Open this publication in new window or tab >>Co-rotating flexible beam with generalized visco-plastic hinges for the nonlinear dynamics of frame structures under impacts
Show others...
(English)Manuscript (preprint) (Other academic)
National Category
Building Technologies
Identifiers
urn:nbn:se:kth:diva-238639 (URN)
Note

QC 20181106

Available from: 2018-11-06 Created: 2018-11-06 Last updated: 2018-12-05Bibliographically approved
4. Energy-conserving scheme of geometrically exact Euler-Bernoulli spatial beam in nonlinear dynamics
Open this publication in new window or tab >>Energy-conserving scheme of geometrically exact Euler-Bernoulli spatial beam in nonlinear dynamics
(English)Manuscript (preprint) (Other academic)
National Category
Building Technologies
Identifiers
urn:nbn:se:kth:diva-238603 (URN)
Note

QC 20181106

Available from: 2018-11-05 Created: 2018-11-05 Last updated: 2018-11-06Bibliographically approved

Open Access in DiVA

fulltext(2090 kB)27 downloads
File information
File name FULLTEXT01.pdfFile size 2090 kBChecksum SHA-512
2faae718452d7dc7ac14dd6f4522a1ae204f20f1ed988f514c5a417686f22656156945b7fa4763b39d8ae8c5e7a5ec3bb87b2214d8861e4c7e06f58f61e8b760
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Chhang, Sophy
By organisation
Structural Engineering and Bridges
Building Technologies

Search outside of DiVA

GoogleGoogle Scholar
Total: 27 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 550 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf