COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2001; 17:385–393 (DOI: 10.1002/cnm.413) Analysis of shear locking in Timoshenko beam elements using the function space approach Somenath Mukherjee 1; *; † and Gangan Prathap 2; ‡ 1 Structures Division; National Aerospace Laboratories; Bangalore 560 017; India 2 Center for Mathematical Modeling and Computer Simulation; Bangalore 560 037; India SUMMARY Elements based purely on completeness and continuity requirements perform erroneously in a certain class of problems. These are called the locking situations, and a variety of phenomena like shear locking, membrane locking, volumetric locking, etc., have been identied. Locking has been eliminated by many techniques, e.g. reduced integration, addition of bubble functions, use of assumed strain approaches, mixed and hybrid approaches, etc. In this paper, we review the eld consistency paradigm using a function space model, and propose a method to identify eld-inconsistent spaces for projections that show locking behaviour. The case of the Timoshenko beam serves as an illustrative example. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS: eld-consistency; function spaces; projection theorems; locking; Timoshenko beam 1. INTRODUCTION Locking is a pathological problem encountered in formulating a certain class of elements for structural analysis, although these elements satisfy completeness and continuity requirements. The problem occurs as shear locking in Timoshenko beams and Mindlin plates, as parasitic shear in two-dimensional elasticity elements, and as membrane locking in arch elements [1]. Various explanations have been proposed for locking. Tessler and Hughes [2] have ar- gued that locking is caused by ill conditioning of the stiness matrix due to the very large magnitude of the shear stiness terms as compared to those of bending stiness. Carpenter et al. [3] have shown that locking occurs due to coupling between the shear deformation and bending deformation, and that it can be eliminated by adopting strain elds such that these are appropriately decoupled. Using the eld consistency paradigm, Prathap [4; 5] has shown that elements lock because they inadvertently enforce spurious constraints that arise from inconsistencies in the strains developed from the assumed displacement functions. ∗ Correspondence to: S. Mukherjee, Structures Division, National Aerospace Laboratories, Bangalore 560 017, India † E-mail: somu@css.cmmacs.ernet.in ‡ E-mail: gp@css.cmmacs.ernet.in Received 27 April 2000 Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 29 January 2001