In this article, several one-dimensional finite elements for the free vibration analysis of composite beams with piezo-electric layers or patches are presented. The displacement components and the electric potential are approximated above the cross section through Lagrange’s polynomials in a layer-wise sense. Thanks to a Unified Formulation, elements’ stiffness and mass matrices are derived in a general, nuclear form that does not depend upon the approximation order of the displacements and the electric potential over the cross section as well as the number of element nodes along the axial direction. Higher-order, layer-wise theories accounting for non-classical effects can be, therefore, straightforwardly formulated. Beam with full piezo-electric layers or piezo-patches are investigated. Results are given in terms of natural frequencies and electro-mechanical coupling coefficients of the bending and torsional modes. Several mechanical and electrical boundary conditions as well as stacking sequences are investigated. Comparison with three-dimensional finite element models is provided showing that the proposed class of finite elements is able to yield accurate results.
AinsworthM (2001) Essential boundary conditions and multi-point constraints in finite element analysis. Computer Methods in Applied Mechanics and Engineering48(190): 6323–6339.
2.
ANSYS (2009) Theory Manual, v12.0. Southpointe, PA: ANSYS, Inc. USA.
3.
BatheKJ (1996) Finite Element Procedure. Prentice Hall.
4.
BenjeddouA (2000) Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Computers & Structures76(1–3): 347–363.
5.
BenjeddouABelouettarS (2006) On the evaluation and application of the modal properties of piezoelectric adaptive structures. In: ToppingBHVMonteroGMontenegroR (eds) Innovation in Computational Structures Technology. Stirlingshire: Saxe-Coburg Publications, pp. 287–302.
6.
BenjeddouADeüJ-FLetombeS (2002) Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation. Thin-Walled Structures40(7–8): 573–593.
7.
BiscaniFGiuntaGBelouettarS. (2011) Variable kinematic beam elements coupled via Arlequin method. Composite Structures93(2): 697–708.
8.
ButzAKlinkelSWagnerW (2008) A geometrically and materially non-linear piezoelectric three-dimensional-beam finite element formulation including warping effects. International Journal for Numerical Methods in Engineering76(5): 601–635.
9.
CarreraE (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering10(3): 215–296.
10.
CarreraEBrischettoSNaliP (2011a) Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis. John Wiley & Sons, UK.
11.
CarreraEGiuntaGPetroloM (2011b) Beam Structures: Classical and Advanced Theories. John Wiley & Sons, UK.
12.
CatapanoAGiuntaGBelouettarS. (2011) Static analysis of laminated beams via a unified formulation. Composite Structures94(1): 75–83.
13.
ChevallierGGhorbelSBenjeddouA (2008) A benchmark for free vibration and effective coupling of thick piezoelectric smart structures. Smart Materials and Structures17(6): 065007/1–11.
DellaCNShuD (2007) Vibration of beams with piezoelectric inclusions. International Journal of Solids and Structures44(7–8): 2509–2522.
16.
D’OttavioMBallhauseDKröplinB. (2006) Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells. Computers & Structures84(22–23): 1506–1518.
17.
D’OttavioMWallmerspergerTKröplinB (2008) Classical and advanced models for laminated plates with piezoelectric layers actuated in shear mode. Mechanics of Advanced Materials and Structures15(3–4): 167–181.
18.
EerNisseEP (1967) Variational method for electroelastic vibration analysis. IEEE Transactions on Sonics and Ultrasonics14(4): 153–160.
19.
El FatmiRGhazouaniN (2011) Higher order composite beam theory built on Saint-Venant’s solution. Part-I: theoretical developments. Composite Structures93(2): 557–566.
20.
GhazouaniNEl FatmiR (2011) Higher order composite beam theory built on Saint-Venant’s solution. Part-II: built-in effects influence on the behavior of end-loaded cantilever beams. Composite Structures93(2): 567–581.
21.
GiuntaGCarreraE (2011) Exact, hierarchical solutions for localized loadings in isotropic, laminated and sandwich shells. Journal of Pressure Vessel Technology: Transactions of the ASME131(4): 041202-1–041202-14.
22.
GiuntaGBelouettarSCarreraE (2010) Analysis of FGM beams by means of classical and advanced theories. Mechanics of Advanced Materials and Structures17(8): 622–635.
23.
GiuntaGBiscaniFBelouettarS. (2011a) Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings. Composite Part B: Engineering42(4): 682–691.
24.
GiuntaGBiscaniFBelouettarS. (2013) Free vibration analysis of composite beams via refined theories. Composite Part B: Engineering44(1): 540–552.
25.
GiuntaGCrisafulliDBelouettarS. (2011b) Hierarchical theories for the free vibration analysis of functionally graded beams. Composite Structures94(1): 68–74.
26.
HernandezVRomanJEVidalV (2005) SLEPc: a scalable and flexible toolkit for the solution of Eigenvalue problems. ACM Transactions on Mathematical Software31(3): 351–362.
27.
KrommerMBerikPVetyukovY. (2012) Piezoelectric shear-response-based torsion actuation mechanism: an exact 3D Saint-Venant type solution. International Journal of Smart and Nano Materials3(2): 82–102.
28.
MilazzoA (2012) An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics. Composite Structures94(6): 2078–2086.
29.
MilazzoAOrlandoC (2012) A beam finite element for magneto-electro-elastic multilayered composite structures. Composite Structures94(12): 3710–3721.
30.
MindlinRD (1952) Forced thickness-shear and flexural vibrations of piezoelectric crystal plates. Journal of Applied Physics23(83): 83–88.
31.
NoorAKBurtonWS (1990) Assessment of computational models for multilayered shells. Applied Mechanics Reviews43(4): 67–97.
32.
PolitOBruantI (2006) Electric potential approximations for an eight node plate finite element. Computers & Structures 84(22–23): 1480–1493.
33.
ReddyJN (2004) Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed.CRC Press, USA.
RoySYuW (2009) A coupled Timoshenko model for smart slender structures. International Journal of Solids and Structures46(13): 2547–2555.
36.
RoySYuWHanD (2007) An asymptotically correct classical model for smart beams. International Journal of Solids and Structures44(25–26): 8424–8439.
37.
SaravanosDAHeyligerPR (1999) Mechanics and computational models for laminated piezoelectric beams, plates and shells. Applied Mechanics Reviews52(10): 305–319.
38.
ShiehRC (1994) Governing equations and finite element models for multiaxial piezoelectric beam sensors/actuators. AIAA Journal32(6): 1250–1258.
39.
TierstenHF (1969) Linear Piezoelectric Plate Vibrations. New York: Plenum Press.
40.
TrindadeMABenjeddouA (2009) Effective electromechanical coupling coefficients of piezoelectric adaptive structures: critical evaluation and optimization. Mechanics of Advanced Materials and Structures16(3): 210–223.
41.
YangLZhifeiS (2009) Free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature. Composite Structures87(3): 257–264.
