Mindlin Plate Element Research Articles

Stress analysis of bimodulus laminates using hybrid stress plate elements

Based on first-order plate theory, a hybrid stress bimodulus Mindlin plate element is developed in this paper. Both the displacement and stress distributions of bimodulus laminated plates are determined. Since the neutral surface deviates from the geometric midplane, an iterative technique is adopted. Compared to the analytical solutions, the present results are better than those, of displacement-based higher-order plate models, although only first-order theory is incorporated herein.

Nonlinear analysis of bonded composite patch repair of cracked aluminum panels

Analyses of adhesively bonded composite patches to repair cracked structures have been the focus of many studies. Most of these studies investigated the damage tolerance of the repaired structure by using linear analysis. This study involves nonlinear analysis of the adhesively bonded composite patch to investigate its effects on the damage tolerance of the repaired structure. The nonlinear analysis utilizes the three-layer technique which includes geometric nonlinearity to account for large displacements of the repaired structure and also material nonlinearity of the adhesive. The three-layer technique uses two-dimensional finite element analysis with Mindlin plate elements to model the cracked plate, adhesive and composite patch. The effects of geometric nonlinearity on the damage tolerance of the cracked plate is investigated by computing the stress intensity factor and fatigue growth rate of the crack in the plate. The adhesive is modeled as a nonlinear material to characterize debond behavior. The elastic-plastic analysis of the adhesive utilizes the extended Drucker-Prager model. A detailed discussion on the effects of nonlinear analysis for a bonded composite patch repair of a cracked aluminum panel is presented in this paper.

Reduced minimization of product and non-product type in Mindlin plate elements

This paper presents concepts of two-dimensional reduced minimization of product type and non-product type: the analytical reduced minimization theory in one dimension when extended to that of Mindlin plate elements reveals that the two-dimensional reduced minimization for Lagrangian plate elements can be obtained by the successive use of the one-dimensional reduced minimization. But this relation does not hold for serendipity elements due to an incompleteness of displacement functions. This paper examines effects of various integrations of shear strain energy by applying the one-dimensional successive reduced minimization to Lagrangian and serendipity plate elements. The results show why the selective 2×3 and 3×2 rule for the transverse shear strain energies of an 8-noded plate element cannot eliminate locking and why a conventional 12-noded plate element of serendipity family does not have one-order-lower, optimal Gauss strain locations. © 1997 John Wiley & Sons, Ltd.

The patch tests and convergence for nonconforming Mindlin plate bending elements

In this paper, the classical Irons\' patch tests which have been generally accepted for the convergence proof of a finite element are performed for Mindlin plate bending elements with a special emphasis on the nonconforming elements. The elements considered are 4-node and 8-node quadrilateral isoparametric elements which have been dominantly used for the analyses of plate bending problems. It was recognized from the patch tests that some nonconforming Mindlin plate elements pass all the cases of patch tests even though nonconforming elements do not preserve conformity. Then, the clues for the Mindlin plate element to pass the Irons\' patch tests are investigated. Also, the convergent characteristics of some nonconforming Mindlin plate elements that do not pass the Irons\' patch tests are examined by weak patch tests. The convergence tests are performed on the benchmark numerical problems for both nonconforming elements which pass the patch tests and which do not. Some conclusions on the relationship between the patch test and convergence of nonconforming Mindlin plate elements are drawn.

Characterization of fatigue crack growth in aluminium panels with a bonded composite patch

This study introduces an analytical procedure to characterize the fatigue crack growth behavior in an aluminium panel repaired with a bonded composite patch. This procedure involves the computation of the stress intensity factor from a two-dimensional finite element method consisting of three layers to model cracked plate, adhesive and composite patch. In this three layer finite element analysis, as recently introduced by the authors, two-dimensional Mindlin plate elements with transverse shear deformation capability are used. The computed stress intensity factor is then compared with the experimental counterpart. The latter was obtained from the measured fatigue crack growth rate of an aluminium panel with a bonded patch by using the power law relationship (Paris Law) of an unpatched aluminum panel. Both a completely bonded patch (with no debond) and a partially bonded patch (with debond) are investigated in this study. This procedure, thus, provides an effective and reliable technique to predict the fatigue life of a repaired structure with a bonded patch, or alternatively, it can be used to design the bonded composite patch configuration to enhance the fatigue life of cracked structure.

Modeling of a cracked metallic structure with bonded composite patch using the three layer technique

Due to the high computational cost of three-dimensional finite element analysis, the two-dimensional finite element analysis involving the three layer technique is introduced to investigate the repair of cracked metallic structures using an adhesively bonded composite patch. In the three layer technique, two-dimensional Mindlin plate elements with transverse shear deformation capability are used for all three layers; cracked plate, adhesive and composite patch. The accuracy of the three layer technique to compute the stress intensity factor for the metallic crack is demonstrated by a comparison with available two- and three-dimensional models. The strain energy release rates of the debond at the adhesive interfaces are also examined and compared with the previous studies. The three layer technique provides an efficient and accurate alternative model which is capable of investigating in depth the adhesive effects on the bonded patch repair of cracked metallic structures.

A Simple Finite Element Formulation for a Laminated Composite Plate with Piezoelectric Layers

A simple finite element formulation for a composite plate with laminated piezoelectric layers is presented. Classical laminate theory with electromechanical induced actuation and variational principles are used to formulate the equations of motion for a Mindlin plate element based upon uniformly reduced numerical integration and hourglass stabilization. The equations of motion are discretized with four-node, 24 degree of freedom quadrilateral shell elements with one electrical degree of freedom per piezoelectric layer. Performance of the finite element formulation developed here has been tested and compared to the experimental and analytical results documented in literature.

Nonlinear FE Analyses of RC Skewed Slab Bridges

Researchers explain the development of a program called NARCOS for the nonlinear finite-element analysis of reinforced concrete skewed slab bridges. The foundation of the program is a layering formulation in which the cross section is divided into steel and concrete layers with nonlinear material properties. Steel layers are simulated with plane stress elements; concrete layers are modeled with four-node plane stress and Mindlin plate elements. The program takes transverse shear deformations into account. Interlayer compatibility is satisfied by constraining in-plane displacements along common interfaces to be the same for adjacent layers. An efficient algorithm is used for assembly of the stiffness matrix and solution of the equilibrium equations. The efficiency of procedures embodied in the program are confirmed through comparison with experimental and other analytical results. Using NARCOS, different models of skewed slab bridges are analyzed in order to assess the relative merits of two different methods of designing such bridges. Bridges designed with increased reinforcement at the obtuse corner have a higher crack-initiation load and a higher ultimate strength than bridges designed with uniform reinforcement in the slab.

Reduced minimization in Lagrangian Mindlin plate element with arbitrary orientation under uniform isoparametric mapping

AbstractFor the displacement‐based Lagrangian Mindlin plate elements oriented arbitrarily under uniform isoparametric mapping without internal distortion, a theoretical interpretation on the conventional shear‐reduced integration is presented by introducing the concept of reduced minimization. It reveals that the conventional shear‐reduced integration such as URI and SRI prevents the unmatched coefficients of shear strains from becoming spurious constraints. Thus, the shear‐reduced integration eliminates spurious constraints and prevents locking.

Reduced minimization of Mindlin plate

AbstractThe use of inconsistent displacement fields for Mindlin plate elements causes unmatched coefficients to appear in shear strain interpolations. The role of the numerical integration order in relation to these unmatched coefficients, and the existence of optimal stress points are explained through the use of the reduced minimization concept. This concept makes it possible to test whether assumed displacement fields or shear strain fields result in inconsistent strain fields which contain spurious constraints. We have included applications of reduced minimization to the conventional C0‐continuous elements, which employ reduced integration, and to quadrilateral Mindlin plate elements of Hinton and Huang to demonstrate how these elements alleviate shear locking.

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

AbstractAn hybrid stress element formulation based on internal, incompatible displacements is used to develop efficient Mindlin plate elements. The 4‐node quadrilateral Mindlin plate element is derived from a modified energy functional. Both displacements and stresses are defined in the natural co‐ordinate interpolation system. The assumed stress field is obtained by tensor transformation and so chosen as to ensure that the element is co‐ordinate invariant and stable. Shear locking is avoided through an appropriate identification of the internal, incompatible displacement field. The role played by incompatible displacements in the formulation of hybrid stress elements for thin and moderately thick plates is discussed. Numerical applications are presented to illustrate the accuracy and reliability of the suggested Mindlin plate element.

On the strain energy release rate for a cracked plate subjected to out-of-plane bending moment

For a through-the-thickness crack in an infinite plate subjected to out-of-plane uniform bending moment, the strain energy release rate is determined using the virtual crack extension and the variation of potential energy. It is shown that the strain energy release rate for the Reissner's plate approaches the classical plate solution as the ratio of plate thickness to crack size becomes infinitesimally small. By using this result, the limiting expression of the stress intensity factor can be explicitly obtained. For general problems, the modified crack closure method is shown to be an efficient tool for evaluating the strain energy release rates from which the stress intensity factor can be calculated. Both the classical plate element and the Mindlin plate element are investigated, and the applicability of the classical plate element is evaluated. Because the stress-free conditions along the crack face lead to inter-penetration of the plate, a line contact model is assumed to investigate the closure effect using Reissner plate theory. Closure at the compressive side is shown to reduce crack opening displacement and consequently the stress intensity factors. When closure is considered, the strain energy rate based on the Reissner plate theory converges to the classical plate solution. This is similar to the nonclosure case.

A general methodology for deriving shear constrained Reissner‐Mindlin plate elements

AbstractIn this paper the necessary requirements for the good behaviour of shear constrained Reissner–Mindlin plate elements for thick and thin plate situations are re‐interpreted and a simple explicit form of the substitute shear strain matrix is obtained. This extends the previous work of the authors presented in References 18 and 31. The general methodology is applied to the re‐formulation of some well known quadrilateral plate elements and some new triangular and quadrilateral plate elements which show promising features. Some examples of the good behaviour of these elements are given.

Vibrations of a Z-cut quartz resonator-structure in the vicinity of the third overtone of extensional mode

Vibrations of a miniature Z-cut, third overtone, extensional quartz resonator were studied. The resonator-structure consisted of a long vibrating beam, a short supporting beam, and a tuning fork. The vibrating beam and supporting beam intersected each other perpendicularly at their middle points. The ends of the supporting beam were in turn connected to the ends of tines of the tuning fork. Two corners near the base of the tuning fork were mounted on supports which had a fixed width but variable lengths. The vibrating beam is mainly excited at the third overtone of extensional mode. The resonator was produced by chemical etching of a Z-cut quartz plate. An anisotropic, four-node quadrilateral, Mindlin plate element model was employed for the study. Incompatible modes were included in the stiffness matrix to prevent shear locking. The frequency spectrum of the resonator as a function of the mounting length and its mode shapes were discussed. The resonator motional resistances for different mounting lengths were measured. Normalized strain energy ratios in the base area and tine area were computed and compared with the normalized motional resistance at various mounting lengths. Good correlations were found: the peaks in strain energy ratio corresponded with the peaks in motional resistance. These peaks occurred at regions in the frequency spectrum which showed strong coupling with other modes of vibration. Large scatterings of the measured motional resistances were also observed at these regions. The strain energy ratio in the base and tines was proposed as a qualitative criterion for evaluating the relative changes in motional resistance of resonators with different mounting lengths and geometries.

A one point integrated assumed strain 4‐node Mindlin plate element

The 4‐node assumed strain elements are among the best elements available today but the bending moments at their full integration points oscillate severely. This paper presents a one point integrated version of the 4‐node assumed strain plate element of Bathe‐Dvorkin. A Taylor series expansion approach is used to accommodate the linear variation of strains/stresses within the element and hence to stabilize the spurious zero energy modes. An extensive number of benchmark results are presented and compared.

A study of stabilization and projection in the 4‐node Mindlin plate element

AbstractIn this paper, a Mindlin plate element is formulated based on the Hellinger–Reissner principle and the γ‐technique. The stiffness consists of a constant stress (one‐point quadrature) matrix and a stabilization matrix. The stabilization matrix is compared with those previously proposed. In addition, the element uses a projection to modify the nodal displacements so that the patch test is satisfied. The projection matrix is based on a mode decomposition. Several numerical cases are presented, and it is shown that the mode decomposition projection is necessary both for satisfaction of the patch test and convergence.

Coupled use of reduced integration and non‐conforming modes in quadratic Mindlin plate element

AbstractThe applications of the reduced integration technique, and the addition of non‐conforming modes and their coupling to Mindlin plate elements to improve their basic behaviour are reviewed and the establishment of a series of new plate elements by combined use of these schemes is presented in this paper. The element formulation is based upon the quadratic Mindlin plate concept. The results obtained by new elements converge to the exact solutions very rapidly as the mesh is refined and show reliable solutions even for severely distorted meshes. The new elements have the requisite numbers of zero eigenvalues associated with rigid body modes to avoid spurious zero energy modes. These elements are shown to solve the shear locking problem completely so that they are applicable to a wide range of plate problems, giving a high accuracy for both thick and thin plates.

Free vibrations of Mindlin plates using the finite element method: Part 1. Square plates with various edge conditions

This paper presents a parametric study of the free vibrations of square plates with various thickness‐to‐span ratios and different boundary conditions. The numerical analysis of these problems has been undertaken using the finite element method. In particular, a nine‐noded, quadrilateral Mindlin plate element has been adopted. The effects of rotatory inertia and shear deformation are included in this model. Results obtained by the finite element analysis are compared with results from various sources. It is suggested that this study might be used to supplement existing benchmark tests.

Displacement and stress predictions from field- and line-consistent versions of the eight-node mindlin plate element

The eight-node isoparametric Mindlin plate bending element based on the serendipity shape functions has a long history of investigation behind it, and has seen various devices to improve it—mixed methods, enforcing of constraints, tensorial transformations, etc. Only very recently have successful versions free of locking in general quadrilateral form and without kinematic modes emerged. In this paper, we shall examine two of the most successful displacement method procedures (a field-consistency approach and a line-consistency approach) and proceeding from these, design three very accurate versions—one based on a variationally correct field-consistency paradigm alone, and two versions derived from the need to ensure consistency of tangential shear strains along principal reference lines so that the usual patch tests are exactly passed. The latter two have shear strain definitions that leave the element free of all problems (locking and kinematic modes) for all boundary suppressions and element distortions whereas the former has two kinematic modes. These line-consistent elements, however, introduce spurious quadratic shear stress oscillations as they have not been derived in a variationally correct sense. The recovery of accurate transverse shear stress resultants must therefore be performed very carefully, and a filtering technique is implemented for this.

Hybrid quadrilateral element based on mindlin/reissner plate theory

Two quadrilateral hybrid Mindlin plate elements Q PL4 and Q PL6 based on a new functional [1] are proposed here. In this formulation the displacements are decomposed into two parts, and the internal displacements are able to impose two different types of constraints on the shear strains of the element. For the first type, locking is eliminated indirectly through increasing the constraints on the shear forces and bending moments, while for the second type the constraints are imposed on the shear strains directly, and corresponds to the technique employed for the displacement type models. It will be shown that the element Q PL6 , which belongs to the first type, is a superior element and can achieve high accuracy and convergence without oscillation over a wide range of plate thickness/span ratios. By adopting the natural coordinate interpolation system, the elements are coordinate invariant. Furthermore, the separation of stress field variables enables the element to yield more accurate stresses. A number of examples is used to demonstrate the accuracy and reliability of the proposed elements.