Higher-order linked interpolation in quadrilateral thick plate finite elements Dragan Ribaric ´ , Gordan Jelenic ´ n University of Rijeka, Faculty of Civil Engineering, V.C. Emina 5, 51000 Rijeka, Republic of Croatia article info Article history: Received 22 January 2011 Received in revised form 30 September 2011 Accepted 22 October 2011 Available online 17 December 2011 Keywords: Finite element method Mindlin plate theory Linked interpolation Higher-order elements abstract In this work, the use of higher-order linked interpolation in the design of plate finite elements is analysed. Benefits of the linked interpolation are well known in the Timoshenko (thick) beam finite elements and in this paper the basis for development of higher-order Mindlin plate elements is found in the analogy between the Timoshenko beam theory and the Mindlin plate theory. The results obtained on standard test examples are compared and numerically assessed against the reference results from literature using various mesh densities and various order of interpolation. & 2011 Elsevier B.V. All rights reserved. 1. Introduction Many finite elements have been developed for the Mindlin moderately thick plates, and in a number of them the idea of linking the displacement field to the rotations of the cross- sections has been thoroughly investigated and exploited [1–16]. As a general conclusion, it has been realised that this idea on its own cannot eliminate the problem of shear locking, especially for coarse meshes, which is in stark contrast to the results obtained by applying the idea to the Timoshenko beam elements [7,17–23]. Consequently, a number of remedies have been proposed, which are mostly based on using adjusted material parameters [24] or on the application of the assumed strain [2,14,15] or the enhanced strain [25] concepts or the use of mixed approaches [1,3,6,11,12] or hybrid approaches [9]. In this paper, we re-visit this classic topic and study the possibility of eliminating the shear-locking problem while remain- ing firmly in the framework of the standard displacement-based finite-element design technique. Within this approach, the kine- matic and constitutive equations of the problem are satisfied in the strong sense, while the equilibrium equations are satisfied in the weak sense with the unknown displacement and rotation fields as the only interpolated quantities. In contrast to most of the existing displacement-based approaches, which base their developments on constraining the shape functions for the displacement and the rotation fields so as to produce a required distribution of the shear strains [8,10,24,26], here we extend the higher-order linked interpolation functions developed for the Timoshenko beam to the plate structures and investigate the results and their relationships with the known approaches. In Section 2, we illustrate the relevant results for the Timoshenko beam elements from [18] giving a family of inter- polation functions which follow a very structured pattern and provide the exact solution for arbitrary polynomial loadings. In some sense, the Mindlin theory of thick plates may be regarded as a 2D generalisation of the Timoshenko theory of thick beams. In stark contrast to thick beams, however, the differential equations of equilibrium for thick plates cannot be solved in terms of a finite number of parameters and consequently no exact finite-element interpolation can be found in this way. Nonetheless, in [1,3,27] such interpolation has been used to formulate three-node trian- gular and four-node quadrilateral thick plate elements, while in [2,6] a six-node triangular and an eight-node quadrilateral elements have been proposed. In Section 3, we firstly consider a quadrilateral four-node plate element, for which the constant shear strain condition imposed on the element edges is known to lead to an interpolation for the displacement field which is dependent not only on the nodal displacements, but also on the nodal rotations around the in- plane normal directions to the element edges. In this way, the displacement interpolation becomes linked to the nodal rotations via shape functions which are linear in one direction and quad- ratic in the other. Such linked interpolation for plates may be also obtained as a 2D generalisation of the linked interpolation for beams [18]. This approach enables an easy generalisation of the linked-interpolation concept to higher-order rectangular plate elements. A different approach to achieve a similar goal has been pursued in [8] where a family of displacement-based Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2011.10.003 n Corresponding author. Tel.: þ385 51 352 114; fax: þ385 51 332 816. E-mail address: gordan@gradri.hr (G. Jelenic ´). Finite Elements in Analysis and Design 51 (2012) 67–80