A finite element model for extension and shear modes of piezo-laminated beams based on von Karman's nonlinear displacement-strain relation


Mechanical Engineering Department, Isfahan University of Technology, Isfahan, 84156-83111, Iran


Piezoelectric actuators and sensors have been broadly used for design of smart structures over the last two decades. Different theoretical assumptions have been considered in order to model these structures by the researchers. In this paper, an enhanced piezolaminated sandwich beam finite element model is presented. The facing layers follow the Euler-Bernoulli assumption while the core layers are modeled with the third-order shear deformation theory (TSDT). To refine the model, the displacement-strain relationships are developed by using von Karman's nonlinear displacement-strain relation. It will be shown that this assumption generates some additional terms on the electric fields and also introduces some electromechanical potential and non-conservative work terms for the extension piezoelectric sub-layers. A variational formulation of the problem is presented. In order to develop an electromechanically coupled finite element model of the extension/shear piezolaminated beam, the electric DoFs as well as the mechanical DoFs are considered. For computing the natural frequencies, the governing equation is linearized around a static equilibrium position. Comparing natural frequencies, the effect of nonlinear terms is studied for some examples


  • An electromechanically coupled finite element model for a piezo-beam was presented.
  • Formulations were developed using the von Karman's displacement-strain relation.
  • The model will be nonlinear if the extension layer is taken as the sensor.
  • The extension asymmetric piezo-sensor causes change in the natural frequencies.


Main Subjects

[1] V.M.F. Correia, M.A.A. Gomes, A. Suleman, C.M.M. Soares, C.A.M. Soares, Modelling and design of adaptive composite structures, Computer Methods in Applied Mechanics and Engineering, 185 (2000) 325-346.
[2] M. Sunar, S.S. Rao, Recent advances in sensing and control of flexible structures via piezoelectric materials technology, Applied Mechanics Reviews, 52 (1999) 1-16.
[3] A. Benjeddou, Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Computers & Structures, 76 (2000) 347-363.
[4] A. Benjeddou, Shear-mode piezoceramic advanced materials and structures: a state of the art, Mechanics of Advanced Materials and Structures, 14 (2007) 263-275.
[5] M.A. Elshafei, F. Alraiess, Modeling and analysis of smart piezoelectric beams using simple higher order shear deformation theory, Smart Materials and Structures, 22 (2013) 035006.
[6] V. Balamurugan, S. Narayanan, A piezoelectric higher-order plate element for the analysis of multi-layer smart composite laminates, Smart Materials and Structures, 16 (2007) 2026.
[7] L.N. Sulbhewar, P. Raveendranath, A novel efficient coupled polynomial field interpolation scheme for higher order piezoelectric extension mode beam finite elements, Smart Materials and Structures, 23 (2014) 025024.
[8] O.J. Aldraihem, A.A. Khdeir, Smart beams with extension and thickness-shear piezoelectric actuators, Smart Materials and Structures, 9 (2000) 1.
[9] M.A. Trindade, A. Benjeddou, On higher-order modelling of smart beams with embedded shear-mode piezoceramic actuators and sensors, Mechanics of Advanced Materials and Structures, 13 (2006) 357-369.
[10] M.A. Trindade, A. Benjeddou, R. Ohayon, Finite element modelling of hybrid active–passive vibration damping of multilayer piezoelectric sandwich beams—part I: Formulation, International Journal for Numerical Methods in Engineering, 51 (2001) 835-854.
[11] A. Benjeddou, M.A. Trindade, R. Ohayon, New shear actuated smart structure beam finite element, AIAA journal, 37 (1999) 378-383.
[12] M.A. Trindade, A. Benjeddou, R. Ohayon, Finite element modelling of hybrid active–passive vibration damping of multilayer piezoelectric sandwich beams—part II: System Analysis, International Journal for Numerical Methods in Engineering, 51 (2001) 855-864.
[13] H. Boudaoud, A. Benjeddou, E.M. Daya, S. Belouettar, Analytical evaluation of the effective emcc of sandwich beams with a shear-mode piezoceramic core, in:  Proceedings of Second International Conference on Desing and Modelling of Mechanical Systems, Monastir, Tunisia, 2007.
[14] J.N. Reddy, A simple higher-order theory for laminated composite plates, Journal of applied mechanics, 51 (1984) 745-752.
[15] M.A. Trindade, A. Benjeddou, Refined sandwich model for the vibration of beams with embedded shear piezoelectric actuators and sensors, Computers & Structures, 86 (2008) 859-869.
[16] M. Krommer, H. Irschik, On the influence of the electric field on free transverse vibrations of smart beams, Smart Materials and Structures, 8 (1999) 401.
[17] P. Muralt, Ferroelectric thin films for micro-sensors and actuators: a review, Journal of Micromechanics and Microengineering, 10 (2000) 136.