Computational Mechanics

, Volume 52, Issue 6, pp 1331–1349 | Cite as

An advanced finite element formulation for piezoelectric beam structures

  • D. LegnerEmail author
  • J. Wackerfuß
  • S. Klinkel
  • W. Wagner
Original Paper


Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.


Piezoelectric beam element Mixed variational formulation 3D constitutive equations Geometric nonlinearity Balanced areas of approximation 



The financial support of the Deutsche Forschungsgemeinschaft for the first author is gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • D. Legner
    • 1
    Email author
  • J. Wackerfuß
    • 2
  • S. Klinkel
    • 3
  • W. Wagner
    • 4
  1. 1.Centre for Research in Computational and Applied Mechanics, Department of Civil EngineeringUniversity of Cape Town Cape TownSouth Africa
  2. 2.Emmy-Noether-Gruppe MISMO, Fachgebiet Festkörpermechanik, Technische Universität DarmstadtDarmstadtGermany
  3. 3.Lehrstuhl für Baustatik und Baudynamik, RWTH Aachen UniversityAachenGermany
  4. 4.Institut für Baustatik, Karlsruher Institut für TechnologieKarlsruheGermany

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