Three‐node assumed strain Mindlin plate finite elements

Abstract

Two new three-node Mindlin plate finite elements are presented, both based on the two-node Timoshenko beam element with problem-dependent cubic linked interpolation. They are developed by generalising a constant shear strain expression of the Timoshenko beam element with rotational fields either derived in a way that makes the whole formulation kinematically consistent, or interpolated independently. Results from the numerical examples indicate robustness and high performance of the presented elements, with no shear locking, which makes them suitable for the analysis of very thin to thick plate cases. Furthermore, one of the presented elements passes the patch test for the constant bending state regardless of the element size. Ultimately, the presented elements are effective and suitable for practical application.

Accurate Numerical Solutions for Standard Skew Plate Benchmark Problems

“…However, the reference values are to a certain extent offset from the ones obtained by various plate finite elements on dense meshes, as it can be observed in Ref. [3]. This is not to be unexpected as the error exists in in both analytical and numerical solutions, and because the reference solutions are given for the limiting thin plate cases only, i.e., shear strains contribution is disregarded.…”

“…The finite element analysis is carried out in the Finite Element Analysis Program (FEAP) [4]. Different three-node shear deformable plate finite elements are used in the analysis, viz., T3-2LIM by Auricchio and Taylor [5] (default plate finite element in FEAP), ARS-T9 by Soh et al [6], T3-U2 by Ribarić and Jelenić [7] and T3-LSI by Grbac and Ribarić [3], so the assessment of the accurate solution is as objective as possible, as some elements, depending on the analysed case, may exhibit better convergence rate than others. The T3-2LIM element is a mixed formulation element with quadratic linked interpolation, in which the rotational fields are enriched with internal degrees of freedom.…”

Hexahedral finite elements with enhanced fixed‐pole interpolation for linear static and vibration analysis of 3D micropolar continuum

“…In the so‐called linked interpolation, the displacement field is interpolated using a one‐degree higher polynomial than the polynomial that interpolates the rotational unknowns, and it has been widely used and thoroughly investigated in finite‐element applications of the Timoshenko beams 33,36,38,39,66 and the Reissner–Mindlin plates 42,54 . The linked interpolation considered here is in its general, problem‐independent, form for a beam with $$$ m $$$ nodes presented in Reference 32 as $$$$ {\mathbf{u}}^h\left({x}_1\right)\kern0.5em =\sum \limits_{i=1}^m{N}_i\left({x}_1\right)\left({\mathbf{u}}_i+\frac{1}{m}\hat{\boldsymbol{\phi} -{\boldsymbol{\phi}}_i}{\mathbf{r}}_{o,i}\right), $$$$ derived from the analytical solution of the differential equation of the spatial Timoshenko beam.…”

“…In the so-called linked interpolation, the displacement field is interpolated using a one-degree higher polynomial than the polynomial that interpolates the rotational unknowns, and it has been widely used and thoroughly investigated in finite-element applications of the Timoshenko beams 33,36,38,39,66 and the Reissner-Mindlin plates. 42,54 The linked interpolation considered here is in its general, problem-independent, form for a beam with m nodes presented in Reference 32 as…”

“…References 32–41) and a sizeable number of attempts have been made to generalise the concept to Mindlin plates (see e.g. References 42–54). When it comes to Cosserats' continua, the use of linked interpolation in static 2D problems has been analysed in Reference 21, but it appears that no such analysis has been performed either for full 3D continuum problems or in dynamics.…”