A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations

Abstract

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.

Error estimates in 2‐node shear‐flexible beam elements

“…It is shown by Prathap [15] that the ÿeld-consistency approach is free from zero energy spurious modes. The force-displacement relationship for the ÿeld consistent ith element having nodes i − 1 and i and subjected uniform transverse pressure is well known (see References [4][5][6]) and can be written as…”

“…Singh and his co-authors [4][5][6] have proposed this new class of ÿnite elements and classiÿed them as material ÿnite elements (MFE). They derived the shape functions of these elements by making use of the equilibrium equations.…”

“…The 2-node beam elements considered herein are; coupled-ÿeld element [4][5][6], ÿeld-consistent element [7,15] and the standard element based on exact integration that locks. The ÿnite element equations, which are class of di erence equations, are expanded into Taylor series about a reference nodal point to obtain di erential equations equivalent to the ÿnite element equations at that node.…”

A mixed stress model for linear elastodynamics of arbitrarily curved beams

“…In these cases, it is necessary to introduce a representation of the centroidal axis. Many papers refer to the circular arch case (see for example, [8][9][10][11]), while the case of arbitrarily curved geometry was afforded earlier using simple straight beams and later using B-splines [12] or cubic Hermite functions. In this work, a modified Hermitian elementwise cubic parametric description is adopted.…”

An Explicit Stiffness Matrix for Parabolic Beam Element

“…Raveendranath and his colleagues (1999) assumed cubic polynomial for radial displacement. By taking advantage of the equilibrium equations, they suggested new displacement functions Raveendranath et al (2001). Similarly, this procedure was extended to create a three-nodded element.…”