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An energy-momentum co-rotational formulation for nonlinear dynamics of planar beams
KTH, Skolan för arkitektur och samhällsbyggnad (ABE), Byggvetenskap. Structural Engineering Research Group/LGCGM, INSA de Rennes, Université Bretagne Loire, 20 avenue des Buttes de Coësmes, CS 70839, 35708 Rennes Cedex 7, France.
KTH, Skolan för arkitektur och samhällsbyggnad (ABE), Byggvetenskap, Byggteknik och design.ORCID-id: 0000-0003-2104-382X
2017 (Engelska)Ingår i: Computers & structures, ISSN 0045-7949, E-ISSN 1879-2243, Vol. 187, s. 50-63Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Elsevier Ltd , 2017. Vol. 187, s. 50-63
Nyckelord [en]
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
Nationell ämneskategori
Samhällsbyggnadsteknik
Identifikatorer
URN: urn:nbn:se:kth:diva-207289DOI: 10.1016/j.compstruc.2017.03.021ISI: 000401675600004Scopus ID: 2-s2.0-85017346411OAI: oai:DiVA.org:kth-207289DiVA, id: diva2:1111823
Anmärkning

QC 20170619

Tillgänglig från: 2017-06-19 Skapad: 2017-06-19 Senast uppdaterad: 2022-06-27Bibliografiskt granskad
Ingår i avhandling
1. Energy-momentum conserving time-stepping algorithms for nonlinear dynamics of planar and spatial Euler-Bernoulli/Timoshenko beams
Öppna denna publikation i ny flik eller fönster >>Energy-momentum conserving time-stepping algorithms for nonlinear dynamics of planar and spatial Euler-Bernoulli/Timoshenko beams
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2018. s. 67
Serie
TRITA-ABE-DLT ; 1836
Nyckelord
Nonlinear Dynamics, Energy-momentum conserving scheme, 2D co-rotational beam, Geometrically exact 3D Euler-Bernoulli beam, impact
Nationell ämneskategori
Husbyggnad
Forskningsämne
Byggvetenskap
Identifikatorer
urn:nbn:se:kth:diva-238598 (URN)978-91-7873-023-0 (ISBN)
Disputation
2018-12-11, Amphi GCU, INSA de Rennes, 20 avenue des Buttes de Coësmes CS 70839 City: Rennes, France, Rennes, 15:00 (Engelska)
Opponent
Handledare
Anmärkning

QC 20181106

Tillgänglig från: 2018-11-06 Skapad: 2018-11-06 Senast uppdaterad: 2022-06-26Bibliografiskt granskad

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Battini, Jean-Marc

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