An accurate coupled field piezoelectric beam finite element formulation is presented. The formulation is based on First-order Shear Deformation Theory (FSDT) with layerwise electric potential. An appropriate through-thickness electric potential distribution is derived using electrostatic equilibrium equations, unlike conventional FSDT based formulations which use assumed independent layerwise linear potential distribution. The derived quadratic potential consists of a coupled term which takes care of induced potential and the associated change in stiffness, without bringing in any additional electrical degrees of freedom. It is shown that the effects of induced potential are significant when piezoelectric material dominates the structure configuration. The accurate results as predicted by a refined 2D simulation are achieved with only single layer modeling of piezolayer by present formulation. It is shown that the conventional formulations require sublayers in modeling, to reproduce the results of similar accuracy. Sublayers add additional degrees of freedom in the conventional formulations and hence increase computational cost. The accuracy of the present formulation has been verified by comparing results obtained from numerical simulation of test problems with those obtained by conventional formulations with sublayers and ANSYS 2D simulations.

An efficient piezoelectric smart beam finite element based on Reddy's third-order displacement field and layerwise linear potential is presented here. The present formulation is based on the coupled polynomial field interpolation of variables, unlike conventional piezoelectric beam formulations that use independent polynomials. Governing equations derived using a variational formulation are used to establish the relationship between field variables. The resulting expressions are used to formulate coupled shape functions. Starting with an assumed cubic polynomial for transverse displacement (w) and a linear polynomial for electric potential (ϕ), coupled polynomials for axial displacement (u) and section rotation (θ) are found. This leads to a coupled quadratic polynomial representation for axial displacement (u) and section rotation (θ ). The formulation allows accommodation of extension-bending, shear-bending and electromechanical couplings at the interpolation level itself, in a variationally consistent manner. The proposed interpolation scheme is shown to eliminate the locking effects exhibited by conventional independent polynomial field interpolations and improve the convergence characteristics of HSDT based piezoelectric beam elements. Also, the present coupled formulation uses only three mechanical degrees of freedom per node, one less than the conventional formulations. Results from numerical test problems prove the accuracy and efficiency of the present formulation.

The present work is devoted to accurate higher-order shear deformation theory (HSDT)-based coupled field finiteelement modelling of piezoelectric extension mode beam. Unlike conventional formulations available in the literature, an accurate through-thickness electric potential distribution consistent with HSDT has been derived from electrostatic equilibrium equations. The derived coupled quartic polynomial field accounts for the induced potential and hence the associated change in stiffness, without introducing any additional electrical degrees of freedom. The parametric studies carried out show the importance of considering induced potential. Conventional independent linear potential formulations available in the literature require a number of sublayers in mathematical modelling of the physical piezoelectric layer to achieve the accurate results, while the present formulation which uses coupled consistent potential efficiently reproduces accurate results, with single-layer modelling of the physical piezoelectric layer. The numerical accuracy and efficiency of the present formulation has been demonstrated by comparing the results obtained for test problems with those obtained by conventional formulations and ANSYS 2D finite-element simulation.

An accurate and efficient coupled polynomial-based interpolation scheme is proposed for the Euler-Bernoulli piezoelectric beam finite element which accommodates induced potential effects and is free from material-locking due to asymmetric distribution of material in the beam cross-section. The consistent through-thickness potential derived from electrostatic equilibrium equation is used, unlike conventional formulations which use assumed linear through-thickness potential. The relationship between mechanical and electrical field variables involved in the formulation is established using governing equations derived from the variational formulation. This relationship is used to derive a coupled polynomial for the axial displacement field with contributions from an assumed cubic polynomial for transverse displacement and linear polynomials for layerwise electric potential. A set of coupled shape functions obtained using these polynomials handles the effects of extension-bending coupling and induced potential in an efficient manner at the field interpolation level itself. The accuracy of the present formulation is proved by comparison of results obtained for test problems with those from ANSYS 2D simulation and conventional formulations. Convergence studies prove the merit of the present coupled polynomial interpolation over the conventional independent polynomial interpolation. This improved performance is achieved with the same number of nodal degrees of freedom as used by conventional formulations.

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Numerical analysis of piezoelectric smart beams plays an important role in the design of smart beam based control systems. In general, smart beams are thin and Euler-Bernoulli piezoelectric beam element is widely used for their structural analysis. Accuracy of Euler-Bernoulli piezoelectric beam element depends on the appropriate assumptions for electric potential involved in the formulation. Most of the Euler-Bernoulli piezoelectric beam finite elements available in the literature assume linear through-thickness potential distribution. It is shown that the accuracy of these conventional formulations varies with relative proportion of piezoelectric material in the total beam cross-section. This is attributed to the effect of the induced potential due to electromechanical coupling. The use of a number of sublayers in the mathematical modeling of each piezoelectric physical layer is shown to improve accuracy, at the cost of additional computational effort due to increased number of nodal electrical degrees of freedom. A more efficient way to handle the effects of induced potential is to use a consistent through-thickness electric potential derived from the electrostatic equilibrium equation. In addition to the conventional linear terms, this field consists of a higher order coupled term which effectively takes care of the geometric effects and produce accurate results, without the use of sublayers in modeling.

Purpose – Piezoelectric extension mode smart beams are vital part of modern control technology and their numerical analysis is an important step in the design process. Finite elements based on First-order Shear Deformation Theory (FSDT) are widely used for their structural analysis. The performance of the conventional FSDT-based two-noded piezoelectric beam formulations with assumed independent linear field interpolations is not impressive due to shear and material locking phenomena. The purpose of this paper is to develop an efficient locking-free FSDT piezoelectric beam element, while maintaining the same number of nodal degrees of freedom. Design/methodology/approach – The governing equations are derived using a variational formulation to establish coupled polynomial field representation for the field variables. Shape functions based on these coupled polynomials are employed here. The proposed formulation eliminates all locking effects by accommodating strain and material couplings into the field interpolation, in a variationally consistent manner. Findings – The present formulation shows improved convergence characteristics over the conventional formulations and proves to be the most efficient way to model extension mode piezoelectric smart beams, as demonstrated by the results obtained for numerical test problems. Originality/value – To the best of the authors’ knowledge, no such FSDT-based finite element with coupled polynomial shape function exists in the literature, which incorporates electromechanical coupling along with bending-extension and bending-shear couplings at the field interpolation level itself. The proposed formulation proves to be the fastest converging FSDT-based extension mode smart beam formulation.

The conventional Timoshenko piezoelectric beam finite elements based on First-order Shear Deformation Theory (FSDT) do not maintain the accuracy and convergence consistently over the applicable range of material and geometric properties. In these elements, the inaccuracy arises due to the induced potential effects in the transverse direction and inefficiency arises due to the use of independently assumed linear polynomial interpolation of the field variables in the longitudinal direction. In this work, a novel FSDT-based piezoelectric beam finite element is proposed which is devoid of these deficiencies. A variational formulation with consistent through-thickness potential is developed. The governing equilibrium equations are used to derive the coupled field relations. These relations are used to develop a polynomial interpolation scheme which properly accommodates the bending-extension, bending-shear and induced potential couplings to produce accurate results in an efficient manner. It is noteworthy that this consistently accurate and efficient beam finite element uses the same nodal variables as of conventional FSDT formulations available in the literature. Comparison of numerical results proves the consistent accuracy and efficiency of the proposed formulation irrespective of geometric and material configurations, unlike the conventional formulations.