An effective quadrilateral plate bending element

An effective arbitrary quadrilateral thin plate bending element with a quasi-conforming, QCQ element, is presented in this paper. The elements pass the patch test with constant strain and the patch test with linear strains approximately. When the element degenerates to a rectangle the patch test with linear strains is passed. The calculation of the element stiffness matrix is simple without numerical integration. The numerical examples show that the QCQ element has a higher accuracy and a faster convergence rate.     

An efficient quadrilateral element for plate bending analysis

A simple, shear flexible, quadrilateral plate element is developed based on the Hellinger/Reissner mixed variational principle with independently assumed displacement and stress fields. The crucial point of the selection of appropriate stress parameters is emphasized in the formulation. For this purpose, a set of guidelines is formulated based on the following considerations: (i) suppression of all kinematic deformation modes, (ii) the element has a favourable value for the constraint index in the thin plate limit, (iii) element properties are frame-invariant. For computer implementation the components of the element stiffness matrix are evaluated analytically using the symbolic manipulation package MACSYMA. The effectiveness and practical usefulness of the proposed element are demonstrated by the numerical results of a variety of problems involving thin and moderately thick plates under different loading and support conditions.     

An efficient hybrid quadrilateral Kirchhoff plate bending element

This paper discusses the formulation of a hybrid stress quadrilateral Kirchhoff plate bending element based on an extended complementary energy functional first proposed by Tong. With the inclusion of a Lagrange multiplier in the functional to enforce a constraint on the assumed moment space, the construction of the C¹ deflection profile inside the element is no longer necessary. The continuity requirement on the deflection across the element interfaces is fulfilled by interpolating the generalized nodal displacements over the element boundary in the usual way. Special attention is paid to the selection of assumed moment space such that the element stability, convergency, invariance and nodal point numbering insensitivity are secured while the implementational cost of the element is kept low. Quadratic serendipity interpolation of the transverse deflection is adopted to discretize the distributed transverse loading. Numerical examples are presented and the accuracy achieved is found to be satisfactory.     

An improved discrete Kirchhoff quadrilateral thin-plate bending element

By combining discrete Kirchhoff theory with least-squares smoothed shape functions, an improved quadrilateral thin-plate bending element is derived. This element exhibits improved performance over an earlier DKQ element in predicting both displacements and stresses. Numerical examples are presented comparing the element with many other quadrilateral elements for standard test problems.     

A formulation for the 4-node quadrilateral element

A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.     

Refined non-conforming quadrilateral thin plate bending element

A refined non-conforming quadrilateral thin plate bending element RPQ4 which can satisfy the requirement of convergence is established such that the non-conforming displacement function can be derived directly. A simple explicit expression of a refined constant strain matrix can be introduced into the formulation of the standard displacement element which results in the constraint condition of interelement continuity being satisfied in an average sense. Numerical examples are presented to show that the present model can pass the patch test and possesses high accuracy. © 1997 John Wiley & Sons, Ltd.     

A new conforming quadrilateral plate bending element

A new four-node conforming quadrilateral element (NCQ) for plate bending is Described. The element is based on an earlier free formulation due to Bergan,^1,2 but extra twist variables are included to make it more accurate. The element geometry is in bilinear polynomials representation, while the displacement functions are described by slight modifications of bicubic polynomials which satisfy energy orthogonality and need not be force-orthogonal. It is shown that the coupling stiffness between the fundamental deformation modes and its higher-order deformation modes has been altered. Numerical examples indicate that the element gives a surprisingly accurate solution.     

A conforming quadrilateral plate bending element revisited

A new formulation of a 16-degree-of-freedom conforming quadrilateral plate bending element using the Cartesian co-ordinates is presented. The displacement function w is taken as a slight modification of bicubic polynomial in the Cartesian co-ordinates. Use of this expression in the construction of finite element representations is expected to be useful specially for microcomputers. The simple stiffness matrix subroutine, coded in FORTRAN 77, for the general quadrilateral plate bending element is listed in the Appendix.     

On the incompressible constraint of the 4-node quadrilateral element

The standard plane 4-node element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the ‘locking’-effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher-order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first-order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty-constraint. Analytical expressions for that constant-dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner. For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the elements are compared.     

Field- and edge-consistency synthesis of A 4-noded quadrilateral plate bending element

In this paper, we demonstrate the use of two conceptual principles, the field-consistency requirement and the edge-consistency requirement, as the basis for deriving a 4-noded quadrilateral plate bending element based on Mindlin plate theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain-interpolation points and Hrennik off strain reference lines, etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element domain, and ‘tangential’ shear strains are consistently matched at element boundaries so that there is no locking even under extreme distortion—e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.     

An improved three-noded triangular element for plate bending

A new three-noded triangular element for plate bending is described. The element is based on an earlier stress-smoothed triangular element due to Razzaque,¹ but extra internal ‘bubble’ functions are included to make it more flexible. The accuracy of the new element is compared with that of a number of other high-performance triangular elements. It is concluded that the present element and that due to Hansen, Bergan and Syvertsen² are the two most accurate triangular thin plate elements currently available. The extra lines of FORTRAN required to convert Razzaque's shape function subroutine to that for the new element are given, thus making the new element easy to implement in any general-purpose finite element program.     

Evaluation of a new quadrilateral thin plate bending element

A review of 4-node, 12 degrees-of-freedom quadrilateral elements for thin plates is presented. A new element called DKQ is discussed. The formulation is based on a generalization of the efficient and reliable triangular element DKT presented in References 1 and 2 and on the rectangular element QC presented in Reference 3. These elements are derived using the so-called discrete Krichhoff technique. A detailed numerical evaluation of the behaviour of the DKQ element for the computation of displacements and stresses for thin plate bending problems is presented and discussed. The DKQ element appears to be a simple and reliable 12 degrees-of-freedom thin plate bending 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 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.     

Linked interpolation for Reissner-Mindlin plate elements: Part III—An alternative quadrilateral

An alternative quadrilateral element for Reissner—Mindlin plates, which extends the mixed formulation described in Part 1 using linked interpolations, is presented. Differently from element Q4BL presented in Part I, this element (Q4BLa) employs four internal rotation nodes and a linear approximation for transverse shear forces. It is shown that the element passes various patch tests, it does not possess any zero-energy mode and its performance is good for both thick and thin plates.     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

A general quadrilateral plate element

The stiffness matrix and consistent nodal load vector for a general quadrilateral plate element are developed for both in-plane and bending analysis. The basis chosen for the development is the hybrid stress method because

• i it has been shown to produce results of relatively high accuracy.
• ii it employs only primary nodes and
• iii all integration required in the derivations can be achieved in closed form.

Various numerical examples, relating to both plane stress and bending conditions, are considered in order to establish the validity and generality of the derivations.     

A new element used in the non-orthogonal boundary plate bending theory—an arbitrarily quadrilateral element

This paper develops an arbitrarily quadrilateral element to analyse bending problems of plates with non-orthogonal boundaries. A second-order Jacobian matrix for the co-ordinate transformation and an explicit form of its inverse matrix are described in detail. A shape function matrix [ N ] for the plate element of arbitrarily quadrilateral configuration, an equivalent load vector { R }, a strain matrix [ B ] and element stiffness matrix are given. Finally, four illustrated examples are given and the results of computation are compared with those from other analytical methods.     

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

In 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.