ARTICLE IN PRESS Finite Elements in Analysis and Design ( ) – www.elsevier.com/locate/finel The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis Charles E. Augarde a , ∗ , Andrew J. Deeks b a School of Engineering, Durham University, South Road, Durham DH1 3LE, UK b School of Civil & Resource Engineering, The University of Western Australia, Crawley Western Australia 6009, Australia Received 29 May 2007; received in revised form 17 January 2008; accepted 18 January 2008 Abstract The exact solution for the deflection and stresses in an end-loaded cantilever is widely used to demonstrate the capabilities of adaptive procedures, in finite elements, meshless methods and other numerical techniques. In many cases, however, the boundary conditions necessary to match the exact solution are not followed. Attempts to draw conclusions as to the effectivity of adaptive procedures is therefore compromised. In fact, the exact solution is unsuitable as a test problem for adaptive procedures as the perfect refined mesh is uniform. In this paper we discuss this problem, highlighting some errors that arise if boundary conditions are not matched exactly to the exact solution, and make comparisons with a more realistic model of a cantilever. Implications for code verification are also discussed.  2008 Elsevier B.V. All rights reserved. Keywords: Adaptivity; Finite element method; Meshless; Closed form solution; Beam; Error estimation; Meshfree 1. Introduction Adaptive methods are well-established for analysis of elas- tostatic problems using finite elements and are now emerging for meshless methods. Many publications in this area measure the capability of adaptive procedures by comparison with the limited number of exact solutions which exist. One of these problems is that of a cantilever subjected to end loading [1]. The purpose of this paper is to highlight potential sources of error in the use of this solution relating to the particular bound- ary conditions assumed and to show that it is a solution nei- ther appropriate for testing adaptivity nor as a model of a real cantilever. While some may consider that the observations we make are self-evident and well-known, the literature contains many counter examples. This paper provides graphic illustration of the effect of various boundary conditions on the cantilever ∗ Corresponding author. Tel.: +44 191 334 2504; fax: +44 191 334 2407. E-mail addresses: charles.augarde@dur.ac.uk (C.E. Augarde), deeks@civil.uwa.edu.au (A.J. Deeks). 0168-874X/$ - see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2008.01.010 beam solution. To our knowledge these effects have not been presented in detail in the existing literature. We also demon- strate the difference between the behaviour of a real cantilever and the idealised Timoshenko cantilever. It is our hope that this paper will help to reduce the misuse of the Timoshenko can- tilever beam in the evaluation of adaptive analysis schemes, and perhaps encourage the use of a more realistic cantilever beam model as a benchmark problem instead. 2. Problem definition Fig. 1 shows a cantilever beam of depth D, length L and unit thickness, which is fully fixed to a support at x = 0 and carries an end load P. Timoshenko and Goodier [1] show that the stress field in the cantilever is given by  xx = P (L - x)y I , (1)  yy = 0, (2)  xy =- P 2I  D 2 4 - y 2  (3) Please cite this article as: C.E. Augarde, A.J. Deeks, The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis, Finite Elem. Anal. Des. (2008), doi: 10.1016/j.finel.2008.01.010