Reissner-Mindlin Model Research Articles

Plate bending analysis using the classical or the Reissner–Mindlin models

Plates can be solved with the classical or the Reissner–Mindlin plate model using the same computer code with an appropriate treatment of the direct boundary element formulation. A field decomposition is employed to establish a connection between these plate models and to obtain a boundary element formulation for the classical model from that used for the Reissner–Mindlin one. The classical model is related to the irrotational component of the field related to the rotations of the plate. The use of this field component in both the fundamental solution and the direct boundary integral equation of the Reissner–Mindlin model carries the classical model approach. Furthermore, the well-known fundamental solution derived from the classical model can be employed as the irrotational field of the fundamental solution of the Reissner–Mindlin model. In this way, the fundamental solution used by Danson to perform classical analysis was used as the irrotational component and the obtained results were compared with those obtained with the well-known Weeën's formulation.

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.

Eigen‐frequencies in thin elastic 3‐D domains and Reissner–Mindlin plate models

AbstractThe eigen‐frequencies of elastic three‐dimensional thin plates are addressed and compared to the eigen‐frequencies of two‐dimensional Reissner–Mindlin plate models obtained by dimension reduction. The qualitative mathematical analysis is supported by quantitative numerical data obtained by the p‐version finite element method.The mathematical analysis establishes an asymptotic expansion for the eigen‐frequencies in power series of the thickness parameter. Such results are new for orthotropic materials and for the Reissner–Mindlin model. The 3‐D and R–M asymptotics have a common first term but differ in their second terms.Numerical experiments for clamped plates show that for isotropic materials and relatively thin plates the Reissner–Mindlin eigen‐frequencies provide a good approximation to the three‐dimensional eigen‐frequencies. However, for some anisotropic materials this is no longer the case, and relative errors of the order of 30 per cent are obtained even for relatively thin plates. Moreover, we showed that no shear correction factor is known to be optimal in the sense that it provides the best approximation of the R–M eigen‐frequencies to their 3‐D counterparts uniformly (for all relevant thicknesses range). Copyright © 2002 John Wiley & Sons, Ltd.

Mixed mode delamination in plates: a refined approach

Abstract Energy release rate and its mode partition in layered plates are analysed by using an improved laminated plate model. The adhesion between layers is modelled by means of a linear interface acting in the opening and sliding failure mode directions. Stress singularities at the crack tip are recovered when the stiffness of the interface approaches infinity. Kirchhoff or Reissner–Mindlin plate models are employed to describe the layers. Analytical solutions of the relevant governing equations are obtained through a variational formulation of an augmented total potential energy, in which the stiffness of the interface introduces kinematics constraints in the form of a penalty functional. Closed form solutions for energy release rates are given evidencing the effectiveness and the simplicity of the proposed model. Comparisons with fracture mechanics results––when available––are shown discussing the validity of the proposed mechanical model to predict mode partition. Interesting features emerging with the introduction of the layer-wise Reissner–Mindlin model are also highlighted, particularly with reference to coupling terms arising from shear effects.

A study of some triangular discrete Reissner–Mindlin plate and shell elements

A shell simultaneously supports membrane and plate bending actions, and both shear and membrane locking may occur. If the element is plane, bending and membrane effects are uncoupled, and the shell element is formed by the assembly of plate and membrane elements. This was proposed in the early sixties, but its success was hampered by the lack of good triangular plate elements. Shear locking has been the major problem, and it may be eliminated by using discrete Reissner–Mindlin models. In this paper, we used some discrete triangular plate elements to form flat shells. Membrane and shear locking are studied.

High-order triangular sandwich plate finite element for linear and non-linear analyses

The objective of this paper is to present a new C 1 six-node triangular finite element for geometrically linear and non-linear elastic multilayered composite plates. This finite element must be able to model both thin and moderately thick plates without any pathologies of the classic plate finite elements (shear locking, spurious modes, …). It is based on a new kind of kinematics proposed by Touratier [1], and built on the Argyris interpolation for bending and the Ganev interpolation for membrane displacements and transverse shear rotations. This kinematics allows to exactly ensure, both the continuity conditions for displacements and transverse shear stresses at the interfaces between layers of a laminated structure, and the boundary conditions at the upper and lower surfaces of the plates. The representation of the transverse shear strains by cosine functions allows avoiding shear correction factors. In addition, transverse normal stress can be deduced with good precision from equilibrium equations at the post-processing level, using the high degree of interpolation polynomia. The element performances are evaluated on some standard plate tests and also in comparison with exact three-dimensional solutions for multilayered plates in linear and non-linear (moderately large deflection) statics, dynamics and for buckling. Comparisons with other plate models using the present finite element approximation and an eight-node C 0 finite element based on the Reissner–Mindlin model are also given. All results indicate that the present element has very fast convergence properties and also gives very accurate results for displacements and stresses.

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.

A rectangular element for the Reissner-Mindlin plate

We investigate a new rectangular element for the Reissner-Mindlin model based on the primitive variable system. Nonconforming rotated Q1 element is used to approximate the transverse displacement and the biquadratic element is used for the rotation. Convergent error estimate is obtained which is independent of the thickness of the plate. AMS(MOS) subject classifications. 65F10, 65F30

Least-squares finite element approximations for the Reissner-Mindlin plate

Based on the Helmholtz decomposition of the transverse shear strain, Brezzi and Fortin in [7] introduced a three-stage algorithm for approximating the Reissner-Mindlin plate model with clamped boundary conditions and established uniform error estimates in the plate thickness. The first and third stages involve approximating two simple Poisson equations and the second stage approximates a perturbed Stokes equation. Instead of using the mixed finite element method which is subject to the 'infsup' condition, we consider a least-squares finite element approximation to such a perturbed Stokes equation. By introducing a new independent vector variable and associated div equation, we are able to establish the ellipticity and continuity of the homogeneous least-squares functional in an H 1 product norm appropriately weighted by the thickness. This immediately yields optimal discretization error estimates for finite element spaces in this norm which are uniform in the thickness. We show that the resulting algebraic equations can be uniformly well preconditioned by well- known techniques in the thickness. The Reissner-Mindlin model with pure traction boundary condition is also studied. Finally, we consider an alternative least-squares formulation for the perturbed Stokes equation by introducing an independent scalar variable.

A two-dimensional model for linear elastic thick shells

In this paper, a two-dimensional model for linear elastic thick shells is deduced from the three-dimensional problem of a shell thickness 2ε, ε > 0. From different scalings on the tangent and normal components of the displacement u ε as widely used in recent works, the limit displacement appears to be Kirchhoff–Love displacement of a different type. It contains additional terms to those found in the Reissner–Mindlin model and satisfies more general equations containing the classical terms found in the literature and some new terms related to the third fundamental form. Such terms could not be well handled in the usual framework. Shear stresses across the thickness are also computed. This model appears to be appropriate to handle stiffened shells which, in fact, cannot be considered uniformly as shallow shells. As a by-product it also lays the mathematical background to justify the Reissner–Mindlin model for plates and will probably contribute to a better understanding of the locking phenomenon encountered in computational mechanics.

On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model

In the framework of thin linear elastic plates it is known that the solutions of both the three-dimensional problem and the Reissner-Mindlin plate model can be developed into asymptotic expansions. By comparing the particular asymptotic expansions with respect to the half-thickness ɛ of the plate in the case of periodic boundary conditions on the lateral side, the shear correction factor in the Reissner-Mindlin plate model can be determined in such a way that this model approximates the three-dimensional solution with one order of the plate thickness better than the classical Kirchhoff model. This fails for hard clamped lateral boundary conditions so that the Reissner-Mindlin model is in this case asymptotically as good as the Kirchhoff model.

A refined multilayered finite-element model applied to linear and non-linear analysis of sandwich plates

This paper presents a computationally efficient and mechanically accurate finite-element formulation for linear and non-linear analysis of flat sandwich panels. Computational efficiency is guaranteed by preserving C 0-continuity as in the Reissner–Mindlin model. Mechanical accuracy is acquired by allowing a zig-zag in-plane displacement field in the thickness direction and by fulfilling interlaminar equilibrium at the interface between the core and skins for the transverse shear stress components. The resulting mixed model is reduced to the standard displacement formulation by writing a weak form of Hooke’s Law. The model is denoted by the acronym RMZC, Reissner–Mindlin, zig-zag, interlaminar continuity. The principle of virtual displacement is employed to write finite element equations governing the linear and non-linear static response, free vibration, buckling and postbuckling behaviour. Non-linear equations are solved iteratively by Newton–Raphson linearization accounting arc-length type strategies. The numerical investigation assesses the proposed model by comparing it to classical lamination theory (CLT), First-order shear deformation theory (FSDT) and other available results. The efficiency and accuracy of the RMZC model is then demonstrated by solving several problems dealing with linear and non-linear bending, natural frequencies, in-plane buckling and postbuckling. Appendices quote details of the most interesting points of the models presented.

Asymptotic consistency of the polynomial approximation in the linearized plate theory

We establish a partial link between two standard methods for deriving plate models from linearized three-dimensional elasticity: the asymptotic method, known to justify the Kirchhoff-Love model, and the polynomial reduction method. In the polynomial method, the reduced model is obtained by projecting the three-dimensional displacement on a closed subspace of admissible displacements, namely displacements that are polynomial with respect to the thickness variable. Our procedure characterizes minimal polynomial subspaces that are consistent with the Kirchhoff-Love model. In the same time, if a singular perturbation term is dropped in the equations of the lower degree model, we recover a Reissner-Mindlin model.

Weight minimization of elastic plates using Reissner-Mindlin model and mixed-interpolated elements

Weight minimization of elastic plates using Reissner-Mindlin model and mixed-interpolated elements

Reissner-Mindlin model for plates of variable thickness. Solution by mixed-interpolated elements

Hard clamped and hard simply supported elastic plate is considered. The mixed finite element analysis combined with some interpolation, proposed by Brezzi, Fortin and Stenberg, is extended to the case of variable thickness and anisotropic material.

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.

Locking Effects in the Finite Element Approximation of Plate Models

We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e., for which the asymptotic convergence rate deteriorates as $d \to 0$, and also show that the p version is free of locking. In order to isolate locking effects from boundary layer effects (which also arise as $d \to 0$), our analysis is carried out for the periodic case, which is free of boundary layers. We analyze in detail the lowest model of the hierarchy, the well-known Reissner-Mindlin model, and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the Riessner-Mindlin case.

Boundary layers of hierarchical beam and plate models

The boundary behavior of a family of hierarchical models of linearly elastic, isotropic plates is studied. The hierarchical models are obtained by spectral semidiscretization of the displacement fields in the transverse direction and strain energy projection. The well known Reissner-Mindlin model is contained in the hierarchy as a special case. A decomposition of the boundary layers of any model in the hierarchy into a bending and a torsion layer, both of which are model dependent, is given. It is shown further that the bending and torsion layers arise as Galerkin approximations of certain (nonlinear) eigenvalue problems in the plate cross section with the subspaces used to derive the hierarchical model. It is shown that the bending layers converge, as the order of the plate model tends to infinity, to the so-called Papkovich functions on an elastic strip. The regularity of the solution on polygonal plates is investigated for the whole hierarchy of plate models and shown to equal the regularity of the plane elasticity problem.

Locking effects in the finite element approximation of plate models

We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e., for which the asymptotic convergence rate deteriorates as d → 0 d \to 0 , and also show that the p version is free of locking. In order to isolate locking effects from boundary layer effects (which also arise as d → 0 d \to 0 ), our analysis is carried out for the periodic case, which is free of boundary layers. We analyze in detail the lowest model of the hierarchy, the well-known Reissner-Mindlin model, and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the Riessner-Mindlin case.

Adaptive finite element analysis of geometrically non‐linear plates and shells, especially buckling

AbstractBased on both moderate and finite rotation bending theories of thin elastic shells including shear deformation, adaptive non‐linear static finite element analysis is treated within a displacement approach and h‐adaptivity. The a posteriori error indicator given by Rheinboldt, gained by linearization, is investigated in order to decide whether the deformations influence the indicator explicitely and how parameter dependent problems (like the Reissner–Mindlin model) behave in the process of adaptation. In order to achieve overall consistency, dimensional adaptivity (to 3‐D elasticity) is implemented within disturbed subdomains, especially at supports.Results are that Rheinboldt's error indicator is valid under certain restrictions but not directly at bifurcation points and that robustness is not improved by adaptation.Nested quadrilateral finite elements are used for studying pre‐ and post‐buckling states of plates and shells.