P. Raveendranath

A two-noded locking–free shear flexible curved beam 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 (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

Free vibration of arches using a curved beam element based on a coupled polynomial displacement field

Abstract The performance of a curved beam finite element with coupled polynomial distributions for normal displacement ( w ) and tangential displacement ( u ) is investigated for in-plane flexural vibration of arches. A quartic polynomial distribution for u is derived from an assumed cubic polynomial field for w using force–moment equilibrium equations. The coupling of these displacement fields makes it possible to express the strain field in terms of only six generalized degrees of freedom leading to a simple two-node element with three degrees of freedom per node. Numerical performance of the element is compared with that of the other curved beam elements based on independently assumed field polynomials. The formulation is shown to be free from any spurious constraints in the limit of inextensional flexural vibration modes and hence does not exhibit membrane locking. The resulting well-conditioned stiffness matrix with consistent mass matrix shows excellent convergence of natural frequencies even for very thin deep arches and higher vibrational modes. The accuracy of the element for extensional flexural motion is also demonstrated.

Application of coupled polynomial displacement fields to laminated beam elements

Abstract A 2-noded curved composite beam element with three degrees of freedom per node is proposed for the analysis of laminated beam structures. The formulation accounts for flexural, extensional and transverse shear loadings in the plane of the curved beam. The transverse shear flexibility based on first-order shear deformation theory is incorporated. A cubic polynomial is assumed for the transverse displacement w. The field interpolations for the longitudinal displacement u and section rotation θ are derived using the elemental equilibrium equations. The procedure leads to field interpolations that are coupled by means of coefficients, which are functions of geometrical and material properties of the element. The efficacy of these coupled polynomial fields in improving the accuracy and convergence characteristics of the proposed element has been demonstrated by a series of numerical examples. The lay-up sequence does not affect the accuracy of the element, unlike the conventional 2-noded elements, which make use of independent field interpolations. The element does not exhibit membrane and shear locking. The test problems prove the versatility of the element for the analysis of curved and straight laminated beams.

A two‐node curved axisymmetric shell element based on coupled displacement field

An efficient two-node curved axisymmetric shell element is proposed. The element with three degrees of freedom per node accounts for the transverse shear flexibility and rotary inertia. The strain components are defined in a curvilinear co-ordinate frame. The variation of normal displacement (w) along the meridian is represented by a cubic polynomial. The relevant constitutive relations and the differential equations of equilibrium in the meridional plane of the shell are used to derive the polynomial field for the tangential displacement (u) and section rotation (θ). This results in interdependent polynomials for the field variables w, u and θ, whose coefficients are coupled by generalized degrees of freedom and geometric and material properties of the element. These coupled polynomials lead to consistently vanishing coefficients for the membrane and transverse shear strain fields even in the limit of extreme thinness, without producing any spurious constraints. Thus the element is devoid of membrane and shear locking in thin limit of inextensible and shearless bending, respectively. Full Gaussian integration rules are employed for evaluating stiffness marix, consistent load vector and consistent mass matrix. Numerical results are presented for axisymmetric deep/shallow shells having curved/straight meridional geometries for static and free vibration analyses. The accuracy and convergence characteristics of this C0 element are superior to other elements of the same class. The performance of the element demonstrates its applicability over a wide range of axisymmetric shell configurations. Copyright

A novel efficient coupled polynomial field interpolation scheme for higher order piezoelectric extension mode beam finite elements

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.

An accurate novel coupled field Timoshenko piezoelectric beam finite element with induced potential effects

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.

A numerically accurate and efficient coupled polynomial field interpolation for Euler–Bernoulli piezoelectric beam finite element with induced potential effect

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.

An accurate higher-order modelling of extension mode smart beams with consistent through-thickness electric potential distribution

The present work is devoted to accurate higher-order shear deformation theory (HSDT)-based coupled field finite-element 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.

A locking-free coupled polynomial Timoshenko piezoelectric beam finite element

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 inter...

A Timoshenko Piezoelectric Beam Finite Element with Consistent Performance Irrespective of Geometric and Material Configurations

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.