Mindlin Plate Bending Element Research Articles

Modelling of Steel-Concrete Composite Structures in Fire Using OpenSees

This paper presents the extension of the structural analysis software framework OpenSees for modeling steel framed composite structures subjected to fire including the development of a geometrically nonlinear shell element. The new shell element is formed by a combination of membrane elements and Mindlin plate bending elements using a general total Lagrangian formulation. The MITC technique (Mixed Interpolation of Tensorial Components) is applied to alleviate shear locking problems and the addition of drilling degrees of freedom is included. A new thermal load class was created to define the temperature distribution through the thickness of the shell section. The two-dimensional OpenSees material, DruckerPrager, was modified to model the concrete in the composite deck slab at elevated temperature with temperature-dependent material properties according to the Eurocode 2. A three-dimensional finite element model of a composite structure was built in OpenSees, consisting of a flat reinforced concrete slab modeled by the developed shell element as well as concrete ribs and beams/columns modeled by three-dimensional beam elements. These components were connected by rigid link elements to model composite action. The performance of the developed model is verified and validated by a series of analytical solutions and experimental results respectively. Among these are: one-way bending of steel plates; fire tests on simply supported composite beams; and reinforced concrete slabs where membrane actions are investigated. Cardington restrained beam test and British Steel Corner test are also modeled. The reasonable agreement achieved between OpenSees predictions and experimental measurements shows the validity of the developed OpenSees extension to model composite structures in fire. The horizontal displacement of the column at floor level was modeled for the first time with reasonable agreement. This work is part of a wider project which, upon completion, will provide a user-friendly open-source computational platform for structural fire engineering analyses from fire dynamic simulation through to heat transfer analysis and mechanical analysis.

Development of a New Cracked Mindlin Plate Element

This work addresses the development of a new four-noded rectangular Mindlin plate bending element (MP4C) with a crack which consists of three degrees of freedom (DOF) at each corner node. The crack in the element is assumed to be not closed and nonpropagating. The crack affects the elastic strain energy and the flexibility matrix of the element, whereas the mass matrix remains unchanged. The complete element stiffness matrix is constructed as the inverse of the combined flexibility matrix of both noncracked and cracked elements. To evaluate the behavior of the proposed cracked Mindlin plate element, numerical examples are provided. They are based on developing user subroutines in ABAQUS. The finite element analysis results using the developed plate element are in excellent agreement with those reported in previous work. The cracked plate element developed in this paper provides a simple and robust approach to model the real service conditions in plate-like structures.

Progressive failure analysis of composite laminated stiffened plates using the finite strip method

A finite strip method for non-linear static analysis based on the tangential stiffness matrix has been developed using the new concept of polynomial finite strip elements, with Mindlin (first-order shear deformable element) plate-bending theory for composite plates. A progressive failure algorithm for composite laminates has been successfully developed for the new finite strip methods using a stress-based failure criterion, Tsai–Wu. A finite strip analysis programming package which is capable of performing non-linear progressive damage analysis for composite stiffened plates and shells has also been developed with Mindlin plate-bending element. Good agreement with the finite element results has been observed through various test cases, confirming the accuracy and reliability of the new developed method.

Improved Modeling Capabilites with Reduced-Order Integration

Many rectangular isoparametric finite elements contain a modeling error known as parasitic shear. Parasitic shear can be removed by applying the correct reduced-order Gauss quardrature integration rule. When the 1×1 rule is blindly applied another error, spurious zero energy modes, can be introduced. An analysis procedure is presented that identifies the correct Gauss integration rule for eliminating parasitic shear without introducing spurious zero energy modes. Displacement polynomials expressed in transparent notation assists in the identification of error sources. The procedure is developed with the four-node membrane element and then applied to analyze and correct four- and nine-node Mindlin plate bending elements. Eigenvalue analyses and sample problems are presented to demonstrate the improvements produced by the procedure.

Non-conforming modes for improvement of finite element performance

This paper presents an efficiency of various non-conforming (NC) modes in development of a series of new finite elements with the special emphasis on 4-node quadrilateral elements. The NC modes have been used as a key scheme to improve the behaviors of various types of new finite elements, i.e., Mindlin plate bending elements, membrane elements with drilling degrees of freedom, flat shell elements. The NC modes are classified into three groups according to the \'correction constants\' of \'Direct Modification Method\'. The first group is \'basic NC modes\', which have been widely used by a number of researchers in the finite element communities. The basic NC modes are effective to improve the behaviors of regular shaped elements. The second group is \'hierarchical NC modes\' which improve the behaviors of distorted elements effectively. The last group is \'higher order NC modes\' which improve the behaviors of plate-bending elements. When the basic NC modes are combined with hierarchical or higher order NC modes, the elements become insensitive to mesh distortions. When the membrane component of a flat shell has \'hierarchical NC modes\', the membrane locking can be suppressed. A number of numerical tests are carried out to show the positive effect of aforementioned various NC modes incorporated into various types of finite elements.

A new triangular stiffened plate element for laminate analysis

A new stiffened plate element for the analysis of laminated stiffened plates has been presented. The basic plate element is a combination of Allman's plane stress triangular element and discrete Kirchhoff–Mindlin plate bending element. The stiffened plate element has been developed to include the effects of transverse shear. The element can accommodate any number of arbitrarily oriented stiffeners and is completely free from the usual constraints on the mesh division of the plates. The element has no problems associated with shear locking — a phenomenon usually encountered in isoparametric elements. Static and free vibration analyses of laminated stiffened plates have been carried out with this element and the results are compared with those published.

A new finite element for buckling analysis of laminated stiffened plates

A new stiffened plate element for stability analysis of laminated stiffened plates has been presented. The basic plate element is a combination of Allman's plane stress triangular element and a Discrete Kirchhoff–Mindlin plate bending element. The element includes transverse shear effects. The model accommodates any number of arbitrarily oriented stiffeners within the plate element and eliminates constraints on the mesh division of the plate. The element has no problem associated with shear locking – a phenomenon usually encountered in isoparametric elements. The stability analysis of laminated stiffened plates has been carried out under different loading conditions with the present element.

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.

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.

Interactive analysis of a space frame-raft-soil system considering soil nonlinearity

A new approach for the physical and material modelling of a space frame-raft-soil system is presented. The physical modelling consists of a modified Timoshenko beam bending element with six degrees of freedom per node to model the beams and columns of the superstructure, a modified Mindlin's plate bending element with five degrees of freedom per node to represent the structural slabs and raft, and the coupled finite-infinite elements with three degrees of freedom per node to model the soil media. The constitutive modelling involves the use of the hyperbolic model to account for the soil nonlinearity. The applicability of the proposed physical model is demonstrated by analysing a four-storey, five- by three-bay space frame. An attempt is also made to compare the linear and nonlinear interactive behaviour of the space frame-raft-soil system.

The four-node C0 Mindlin plate bending element reformulated, Part I: Formulation

A new formulation was proposed recently for the removal of the locking mechanisms from the C 0 structural elements. The formulation proposed, although exhibiting an excellent performance in removing the locking mechanisms, was also found to yield flexible models in some cases of plate and shell element applications. The analysis of this behaviour revealed that an internal moment redistribution mechanism, present with the classical formulation of the C 0 structural elements, was missing from the new one. Subsequently, a modified formulation was proposed with all locking mechanisms being rejected, while allowing for the internal moment redistribution mechanism to be retained. The present paper deals with the implementation of the modified formulation in the case of the four-node Mindlin plate bending element and the analysis of its behaviour with respect to the moment redistribution mechanism.

The performance of a reformulated four-node plate bending element in moderately thick to very thin plate applications

This paper analyzes the behavior of a reformulated four-node Mindlin plate bending element in thin to very thin and in moderately thick element applications. The shear locking mechanism is rejected with the proposed formulation whereas machine-related locking remains the only problem potentially developed in thin element applications of the reformulated element. The solution to the machine locking problem may be achieved with the implementation of a simple technique rendering the element completely locking-free. In moderately thick element applications, the reformulated four-node plate bending element is shown to exhibit an excellent performance in accordance with Mindlin-Reissner theory. This performance, if coupled with an analogous behavior of the element in modeling boundary layers developing in thin plate applications (a work to appear soon), renders the reformulated four-node Mindlin plate bending element one of the most reliable and robust, yet simple, Mindlin C 0 plate bending elements.

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.

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.