Reissner-Mindlin Plate Bending Research Articles

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

We consider a stabilized finite element formulation for the Reissner–Mindlin plate bending model. The method uses standard basis functions for the deflection and the rotation vector. We apply a standard multigrid algorithm to obtain a preconditioner. We prove that the condition number of the preconditioned system is uniformly bounded with respect to the mesh size and the thickness parameter. In order to apply the abstract multigrid theory, we introduce new interpolation norms and prove error estimates in them. Numerical results confirm the analysis.

Superconvergence and postprocessing of MITC plate elements

In this paper, the MITC finite element methods for the Reissner–Mindlin plate bending problem are considered. A superconvergence result for the deflection is proved. Utilizing this property a local postprocessing is introduced and analyzed. The improved accuracy of the deflection is confirmed by numerical computations.

On the accuracy of Reissner–Mindlin plate model for stress boundary conditions

For a plate subject to stress boundary condition, the deformation determined by the Reissner–Mindlin plate bending model could be bending dominated, transverse shear dominated, or neither (intermediate), depending on the load. We show that the Reissner–Mindlin model has a wider range of applicability than the Kirchhoff–Love model, but it does not always converge to the elasticity theory. In the case of bending domination, both the two models are accurate. In the case of transverse shear domination, the Reissner–Mindlin model is accurate but the Kirchhoff–Love model totally fails. In the intermediate case, while the Kirchhoff–Love model fails, the Reissner–Mindlin solution also has a relative error comparing to the elasticity solution, which does not decrease when the plate thickness tends to zero. We also show that under the conventional definition of the resultant loading functional, the well known shear correction factor 5/6 in the Reissner–Mindlin model should be replaced by 1 . Otherwise, the range of applicability of the Reissner–Mindlin model is not wider than that of Kirchhoff–Love's.

On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models

We show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchho-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear eect, the Kirchho-Love model fails.

On the connection between some linear triangular Reissner-Mindlin plate bending elements

We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence properties.

Error estimation through the constitutive relation for Reissner–Mindlin plate bending finite elements

This paper deals with the so-called ‘error estimation through the constitutive relation’ for Reissner–Mindlin plate bending finite element analysis, which is based on the calculation of statically admissible stress resultant fields. Further an energy norm through the constitutive relation is introduced to quantify the difference between the calculated statically admissible stress resultant fields and the stress resultants given by the finite element analysis. The statically admissible stress resultant fields are obtained by local calculations at the node or at the element level. The procedures involve an optimization for estimating the statically admissible quantities as close as possible to the finite element result, and thus lead to a more accurate error estimation, considering the Pragger–Synge theorem. The set of examples shows the good agreement between the proposed indicator and the error obtained from the analytical solution. It is concluded that plate bending analysis is a field of application which is well fitted for error estimation through the constitutive relation.

On constructing accurate recovered stress fields for the finite element solution of Reissner-Mindlin plate bending problems

The effectiveness and reliability of three local least squares fit procedures for the construction of smoothed moments and shear forces for the solution of the Reissner-Mindlin (RM) thick plate model are reviewed. The three procedures considered are, namely, the superconvergent patch recovery technique (SPR), the recovery by equilibrium in patches (REP) and the recovery procedure suggested by Lee, Park and Lee (LP). Numerical studies are carried out by applying these three recovery procedures for the construction of smoothed stress fields for various RM plate bending problems. The results obtained indicate that if the order of polynomial used in the stress recovery is the same as that for the finite element analysis, all three procedures produce recovered stress fields with very similar accuracy and convergence rate. However, when the order of polynomial is raised, if the exact solution is smooth without any strong boundary layers or singularities, both the REP and LP procedures can produce more accurate stress fields than the SPR. In case that singularities or boundary layers are present, the LP procedure often outperforms the REP procedure and results in more stable recovery matrices and more accurate recovered stress fields.

On evaluating the inf-sup condition for plate bending elements

This paper addresses the evaluation of the inf–sup condition for Reissner–Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed finite element scheme is, in general, very difficult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain specific results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf–sup condition, we find that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. © 1997 John Wiley & Sons, Ltd.

An efficient six-node plate bending hybrid/mixed element based on mindlin/reissner plate theory

A new efficient hybrid/mixed thin~moderately thick plate bending element with 6-node (HM6-14) is formulated based on the Reissner-Mindlin plate bending theory. The convergence of this element is proved by error estimate theories and verified by patch test respectively. Numerical studies on such an element as HM6-14 demonstrate that it has remarkable convergence, invariability to geometric distorted mesh situations, to axial rotations, and to node positions, and no "locking" phenomenon in thin plate limit. The present element is suitable to many kinds of shape and thin~moderately thick plate bending problems. Further, in comparison with original hybrid/mixed plate bending element HP4, the present element yields an improvement of solutions. Therefore, it is an efficient element and suitable for the development of adaptive multi-field finite element method (FEM).

Avoiding locking in Reissner-Mindlin plates with high-order hierarchic bubble plus finite elements

In this paper we prove that special finite elements of suitable degree do not exhibit any kind of locking phenomenon when used for the numerical solution of Reissner-Mindlin plate bending problems. Despite its simple approach the discretization of the Reissner-Mindlin model is not straightforward. The inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects. The approximate solution is very sensitive to the plate thickness and, for small thickness, it is very far from the true solution. The phenomenon is known as locking of the numerical solution. The finite element we present is constructed by adding bubble functions to standard Serendipity elements. The new elements, referred to as bubble plus elements, from the degree three onwards avoid the numerical locking. The mathematical proof of such a good behavior is based on a deep relation between the C 0 bubble plus element of degree three for Reissner-Mindlin plates and the C 1 element for Kirchhoff plates known as Bogner-Fox-Schmit element.

Mesh adaptation using a four-noded quadrilateral plate bending element

This paper uses a four-noded quadrilateral Reissner-Mindlin plate bending element for mesh adaptation. To overcome the problems of “locking” in the thin plate limit and zero energy modes the transverse shear strains are not evaluated from the displacements and rotations but are separately interpolated. The element is used in a hierarchical mesh adaptation which uses a node-averaging-based energy norm error estimator. Several examples illustrate the use of the adaptive algorithm.

A NOTE ON UNIFORM SUPERCONVERGENCE FOR THE TIMOSHENKO BEAM USING MIXED FINITE ELEMENTS

In this paper we present some results on the discretization by mixed finite elements of the Timoshenko beam, i.e. the one-dimensional Reissner-Mindlin plate bending problem. The results concern superconvergence. Superconvergence (of the displacement at nodal points and of the gradient at Gaussian points) for plate bending problems was considered before, but these earlier results degenerate for small values of the plate thickness d. Here, we prove superconvergence of the mixed finite element solutions to projections of the real solutions on the approximating spaces in the global H1(I)-norm uniform in d. These facts can be used to obtain asymptotically exact a posteriori error estimators, uniform in d, by means of an easy implementable and cheap post-processing. Numerical experiments illustrate the conclusions.

Combining hierarchic high order and mixed-interpolated finite elements for Reissner—Mindlin plate problems

The Reissner—Mindlin plate bending model describes the deformation of a plate subject to a transverse loading when transverse shear deformation is taken into account. Despite its simple approach, the discretization of the Reissner—Mindlin model is not straightforward. The inclusion of the transverse shear strain effect in standard finite element models introduces undesirable numerical effects. The approximate solution is very sensitive to the plate thickness and, for small thickness, it is very far from the true solution. The phenomenon is known as locking of the numerical solution. The most common way to avoiding the locking problem is to use non-standard finite elements and/or modify the variational formulation. Recently, numerical experiences with high order finite elements applied to the plain Reissner—Mindlin formulation have shown a consistent improvement in the quality of the results. Meanwhile some mixed-interpolated finite elements have been suggested and shown to be locking free. In this paper we propose the combination of the two classes of elements introducing a family of hierarchic high order mixed-interpolated finite elements.

An adaptive h‐refinement using transition element for plate bending problems

AbstractAn adaptive h‐refinement procedure utilizing compatible 5‐ to 7‐node quadrilateral transition elements together with 4‐node elements is presented for Reissner–Mindlin plate bending problems. By introducing the transition elements which have physical midside nodes, some difficulties associated with imposing displacement constraints on irregular nodes to enforce interelement compatibility in common adaptive h‐refinement on quadrilateral mesh are easily overcome. Numerical examples are presented to show the effectiveness of using the transition elements for the adaptive h‐refinement.

Accurate evaluation of support reactions in finite element plate-bending analysis

Two simple approaches are presented which allow the distribution of support reactions to be predicted with as high degree of accuracy as the displacements. In the first approach the plate element assembly is completed with special one-dimensional elastic support elements. If their Winkler coefficient is suitably tuned, an accurate prediction of reactions is obtained as a part of the finite element analysis without unduly affecting the displacements and moments of the plate. In the second approach, a standard finite element calculation (without elastic support elements) is performed first and the distribution of reactions is then evaluated based on the known nodal forces at boundary nodes of the plate. The two approaches are indiscriminately applicable with Kirchhoff and Reissner-Mindlin plate bending elements. Their practical efficiency is illustrated by numerical examples.

Mixed‐interpolated elements for Reissner–Mindlin plates

AbstractWe present in this paper a procedure to establish Reissner–Mindlin plate bending elements. The procedure is based on the idea to combine known results on the approximation of Stokes problems with known results on the approximation of elliptic problems. The proposed elements satisfy the mathematical conditions of stability and convergence, and some of them promise to provide efficient elements for practical solutions.

Fracture analysis of plate bending using the finite element method

Xthe transverse deflection and rotat~ons Yn the Reissner-Mindlin Sype of plate bending theories, and r is the r adial distance from the crack tip. This result suggests that two different finite element interpolation functions need to be utilizkd for w and 0 respectively, in the application of i' the Reissner-Mindlin plate bending theory to fracture problems. The commonly used quarter-point singular isoparametric elements c annot satisfy the condition stated above, since the same interpolations should be used for w and Bi. Instead, a f amily of singular quadrilateral isoparametric elements derived in [5] c an be a pplied to this case. For example, an eight-noded singular element with derivative singularity of order p is given below in terms of local coordinate axes s and t:

A six‐node finite element for plate bending

AbstractA six‐node plate bending element has been developed by employing mixed formulation based on a modified Hellinger–Reissner principle and the Reissner–Mindlin plate bending theory. The numerical result indicates that, among the types of assumed independent transverse shear strains considered, a combination of 2α version with either 5α version or 6α version is free of spurious kinematic modes and leads to accurate and reliable solutions even for very thin plates.