SUMMARYAn e cient shear-exible three-noded curved beam element is proposed herein. The shear exibility is based on Timoshenko beam theory and the element has three degrees of freedom, viz., tangential displacement (u), radial displacement (w) and the section-rotation (Â). A quartic polynomial interpolation for exural rotation is assumed a priori. Making use of the physical composition of  in terms of and u, a novel way of deriving the polynomial interpolations for u and w is presented, by solving force-moment and moment-shear equilibrium equations simultaneously. The ÿeld interpolation for  is then constructed from that of and u. The procedure leads to high-order polynomial ÿeld interpolations which share some of the generalized degrees of freedom, by means of coe cients involving material and geometric properties of the element. When applied to a straight Euler-Bernoulli beam, all the coupled coe cients vanish and the formulation reduces to classical quintic-in-w and quadratic-in-u element, with u; w, and @w=@x as degrees of freedom. The element is totally devoid of membrane and shear locking phenomena. The formulation presents an e cient utilization of the nine generalized degrees of freedom available for the polynomial interpolation of ÿeld variables for a three-noded curved beam element. Numerical examples on static and free vibration analyses demonstrate the e cacy and locking-free property of the element.

A new two-noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation ( ) are determined by employing force-moment and moment-shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and for curved element configurations, which reduces to linear and quadratic polynomials for u and , respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class.

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A new two‐noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation (θ) are determined by employing force‐moment and moment‐shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and θ for curved element configurations, which reduces to linear and quadratic polynomials for u and θ, respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C0 element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class. Copyright © 1999 John Wiley & Sons, Ltd.

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.