Journal of Elasticity

, Volume 88, Issue 3, pp 223–253 | Cite as

Behavior of Piezoelectric Beams under Axially Non-uniform Distributed Loading

  • Vladimir Rovenski
  • Haim Abramovich


The paper presents a generic solution methodology for a quasi-static homogeneous monoclinic piezoelectric beam under axially distributed electric and mechanical surface loads and body forces expressed as polynomials of degree K≥ 0 of the axis variable. (In the absence of any electrical loading, this problem is known as the Almansi–Michell problem). The stress and the electrical displacement components are presented as a set of polynomials of degree ≤K+2 of the axis variable (“solution hypothesis”) containing 4K unknown tip loading constants and 3K stress functions of two variables. The cases K=0,1 stand for uniform or linear distributed loads in the axis direction. The analysis is initiated by the Kth level and continues down to lower levels. The main result of this work generalizes the “elastic” solution given recently by O. Rand and the first author (2005). Examples of solutions for axially uniform distributed loads (K=0), and equilibrium in which the stress and the electrical displacement do not depend on the axis variable, are presented. The applications to constant body loads and a hydrostatic pressure are considered.


Quasistatic piezoelectric beam Almansi–Michell problem 

Mathematics Subject Classifications (2000)

74D05 74B05 


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  1. 1.
    Almansi, E.: Sopra la deformazione dei cilindri sollecitati lateralmente. Atti della Academia Nazionale dei Lincei, 10, Nota I: 333–338; Nota II: 400–408 (1901)Google Scholar
  2. 2.
    Bors, C.: Almansi–Michell problem for an elastic orthotropic cylinder. Atti R. Accad. Naz. Lincei 54(3), 441–446 (1973) (Roma)Google Scholar
  3. 3.
    Dzhanelidze, G.Yu.: The Almansi problem. In: Proceedings of Leningrad Polytechnic Institute, vol. 210, pp. 25–38, (1960) (in Russian)Google Scholar
  4. 4.
    Iesan, D.: On the theory of uniformly loaded cylinders. J. Elast. 16(4), 375–382 (1989)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Khatiashvili, G.: The Almansi–Michell problems for homogeneous and composite bodies. Metsniereba. Tbilisi pp. 237, I (1983), pp. 185, II (1985) (in Russian)Google Scholar
  6. 6.
    Ruchadze, A., Berekašvili, R.: On a generalized Almansi problem. Soobshch. Akad. Nauk Gruzin. SSR 100(3), 561–564 (1980) (in Russian)MathSciNetGoogle Scholar
  7. 7.
    Wang, M.Z., Xu, X.S.: A generalization of Almansi’s theorem and its application. Appl. Math. Model. 14(5), 275–279 (1990)zbMATHCrossRefGoogle Scholar
  8. 8.
    Varisov, M.-Z.: The Almansi problem for a cylindrically anisotropic body. Trudy Inst. Vychisl. Mat. Akad. Nauk Gruzin. SSR 27(1), 3–11 (1987) (in Russian)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Rovenski, V., Harash, E., Abramovich, H.: St.Venant’s problem for homogeneous piezoelectric beams. J. Appl. Mech. 74, 1–10 (2007)CrossRefGoogle Scholar
  10. 10.
    Batra, R., Yang, J.: Saint-Venant’s principle in linear piezoelectricity. J. Elast. 38(2), 209–218 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kudryavtsev, B.A., Parton, V.Z., Senik, N.A.: Electromagnetoelasticity. Applied Mechanics: soviet reviews, vol. 2, pp. 1–230. Hemisphere Publ. Co. (1990)Google Scholar
  12. 12.
    Rand, O., Rovenski, V.: Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools. Birkhauser, Boston, USA (2005)zbMATHGoogle Scholar
  13. 13.
    St. Venant, B.: Memoire sur la flexion des prismes. J. math, pures et appl. (Liouville). Ser II, 1, 89–189 (1856)Google Scholar
  14. 14.
    St. Venant, B.: Memoire sur la torsion des prismes, Memoires presentes par divers savants a l’academie des sciences. Sciences Math. et Phys. Paris 14, 233–560 (1856)Google Scholar
  15. 15.
    Michell, J.H.: The theory of uniformly loaded beams. J. Math. 32, 28–42 (1901)Google Scholar
  16. 16.
    Cady, W.G.: Piezoelectricity – An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, vol. 1. Dover Publ. New-York (1964)Google Scholar
  17. 17.
    Rovenski, V., Abramovich, H.: Almansi–Michell Problem for Homogeneous Piezoelectric Beams. TAE report No. 969, 1–108 (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Faculty of Aerospace EngineeringTechnion – I.I.T.HaifaIsrael

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