Coupled polynomial field approach for elimination of flexure and torsion locking phenomena in the Timoshenko and Euler–Bernoulli curved beam elements Md. Ishaquddin a , P. Raveendranath b,n , J.N. Reddy c a Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India b Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram 695547, India c Department of Mechanical Engineering, Texas A & M University, College Station, TX 77843-3123, USA article info Article history: Received 16 October 2012 Accepted 19 October 2012 Available online 27 November 2012 Keywords: Finite elements Curved beam Timoshenko beam Euler–Bernoulli beam Shear locking Flexure locking Torsion locking Coupled polynomial field abstract The curvature related locking phenomena in the out-of-plane deformation of Timoshenko and Euler– Bernoulli curved beam elements are demonstrated and a novel approach is proposed to circumvent them. Both flexure and Torsion locking phenomena are noticed in Timoshenko beam and torsion locking phenomenon alone in Euler–Bernoulli beam. Two locking-free curved beam finite element models are developed using coupled polynomial displacement field interpolations to eliminate these locking effects. The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler–Bernoulli beam elements using the governing equations. The presence of penalty terms in the coupled displacement fields incorporates the flexure–torsion coupling and flexure–shear coupling effects in an approximate manner and produce no spurious constraints in the extreme geometric limits of flexure, torsion and shear stiffnesses. The proposed coupled polynomial finite element models, as special cases, reduce to the conventional Timoshenko beam element and Euler–Bernoulli beam element, respectively. These models are shown to perform consistently over a wide range of flexure- to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios and are inherently devoid of flexure, torsion and shear locking phenomena. The efficacy, accuracy and reliability of the proposed models to straight and curved beam applications are demonstrated through numerical examples. & 2012 Elsevier B.V. All rights reserved. 1. Introduction The major deficiency of displacement based beam finite elements is their tendency to exhibit locking phenomena when applied to slender regimes. These locking effects are primarily due to poor kinematic representation of field variables which pollutes the performance of finite elements and leads to delayed conver- gence. It is important to eliminate these ill-effects in engineering practice. During recent years considerable efforts have been made to develop different numerical schemes to eliminate the defi- ciency of locking in beam finite elements. As per the literature survey, the researchers mainly focused on elimination of shear and membrane locking phenomena associated with in-plane deformation of beam finite elements [1–13]. The studies on locking mechanisms associated with out-of-plane behavior of beam element are limited to elimination of shear locking and extraction of stress resultants accurately [14–20]. It is well known that the shear and membrane locking mechanisms fail to repro- duce the true physical behavior of ‘shearless bending’ and ‘inextensible bending’, respectively, in the thin regimes. It was believed that curved beam finite elements exhibit only shear and membrane locking. In contrast, our recent investigation [21] revealed the existence of two new curvature related locking phenomena: flexure locking and torsion locking. These are asso- ciated with out-of-plane deformation of curved beam elements. The severity of these phenomena was shown to depend on the magnitude of flexure-to-torsion (EI/GJ) stiffness ratio. It was demonstrated that, the use of inconsistent torsion strain models produce spurious torsion strain energy and fail to simulate accurately the ‘torsionless flexure’ behavior when GJ*EI, leading to torsion locking. Similarly, the use of inconsistent flexure strain models produce spurious flexure strain energy and fail to simulate accurately the ‘flexureless torsion’ behavior when EI*GJ, leading to flexure locking. In this article, we propose two locking-free finite element models based on coupled polynomial displacement fields to alleviate the curvature related locking phenomena associated Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.10.005 n Corresponding author. E-mail addresses: vu2rvj@yahoo.com, raveendranath@iist.ac.in (P. Raveendranath). Finite Elements in Analysis and Design 65 (2013) 17–31