The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element

The skew Cartesian coordinate system determined by the Jacobian of the isoparametric transformation evaluated at the origin can be shown to be a geodesic coordinate system at the origin. By using a theory in differential geometry, inverse relations of the isoparametric coordinate transformation can be derived and expressed in terms of these geodesic coordinates. In the formulation of hybrid stress finite elements, it is suggested as a new strategy for assumed stresses that such coordinates be used as the reference coordinates. The theory described is exemplified by its applications to the 4-node hybrid stress membrane elements. A set of new distortion-measuring parameters for the quadrilateral element are also proposed based on such theory.     

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

An 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.     

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.     

Four-node quadrilateral stress membrane element with rotational stiffness

A four-node quadrilateral membrane element with 12 degrees-of-freedom and based on stress assumptions is presented. The element has a rotational degree-of-freedom at each node associated with a moment.     

A penalty finite element approach for couple stress elasticity

There is mounting evidence for size dependent elastic deformation at micron and submicron length scales. Material formulations incorporating higher order gradients in displacements have been successful in modeling such size dependent deformation behavior. A couple stress theory without micro-rotation is considered here as micro-rotations increase complexity and necessitate parameters that are difficult to determine. Higher order gradient theories require continuity in both displacements and their derivatives and direct approaches with both displacements and their derivatives as nodal variables results in a large number of degrees of freedom. Here nodal rotations are applied along with nodal displacements to obtain a simpler finite element formulation with fewer degrees of freedom. The difference in rotation gradients determined with nodal displacements and rotations are minimized by a penalty term. To assess the suggested approach simulations are presented and discussed, where the material parameters have been obtained from experiments of epoxy microbeams in the literature.     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

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 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.     

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.     

Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method

The transverse vibration of a rotary tapered microbeam is studied based on a modified couple stress theory and Euler–Bernoulli beam model. The governing differential equation and boundary conditions are derived according to Hamilton's principle. The generalized differential quadrature element method is then used to solve the governing equation for cantilever and propped cantilever boundary conditions. The effect of the small-scale parameter, beam length, rate of cross-section change, hub radius, and nondimensional angular velocity on the vibration behavior of the microbeam is presented.     

The investigation of the nonlocal longitudinal stress waves with modified couple stress theory

In the present work, the propagation of longitudinal stress waves is investigated using a modified couple stress theory. The analysis of wave motion is based on a Love rod model including the effects of lateral deformation. The present analysis also considers the effect of shear stress components. By applying Hamilton??s principle, the explicit nonlocal elasticity solution is obtained, and the effects of shear stress and length scale parameter are discussed.     

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.     

A robust refined quadrilateral plane element

Based on a rational choice of the internal incompatible displacement function and a special formulation of the a priori elimination of the internal non-conforming displacement parameters, a new refined quadrilateral plane element RQ₄ has been developed. The present element can be shown to be computationally efficient, accurate and free from locking, and is better than other elements such as the Plan's element HS, the generalized hybrid element Q_CS6, and the refined hybrid element RGH₄, etc. Several numerical examples are given to show the superior performances of the present element RQ₄.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.     

A robust,four-node,quadrilateral element for stress analysis of functionally graded plates through higher-order theories

ABSTRACT

A hybrid-mixed, four-node, quadrilateral element for the three-dimensional (3D) stress analysis of functionally graded (FG) plates using the method of sampling surfaces (SaS) is developed. The SaS formulation is based on choosing an inside the plate body N, not equally spaced SaS parallel to the middle surface, in order to introduce the displacements of these surfaces as basic plate variables. Such a choice of unknowns, with the consequent use of Lagrange polynomials of the degree N ? 1 in the assumed distributions of displacements, strains, and mechanical properties through the thickness leads to a robust FG plate formulation. All SaS are located at Chebyshev polynomial nodes that permit one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and spurious zero-energy modes, the assumed natural strain method is employed. The proposed four-node quadrilateral element passes 3D patch tests for FG plates and exhibits a superior performance in the case of coarse distorted meshes. It can be useful for the 3D stress analysis of thin and thick metal/ceramic plates because the SaS formulation gives an opportunity to obtain the solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity.     

A novel versatile multilayer hybrid stress solid-shell element

This paper presents a versatile multilayer locking free hybrid stress solid-shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, i.e. from shell-like isotropic structures to multilayer anisotropic composites. This solid-shell element has eight nodes with only displacement degrees of freedom and a few internal parameters that provide the locking free behavior and accurate interlaminar stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well-known Fraeijs de Veubeke–Hu–Washizu (FHW) multifield variational principle with enhanced assumed strains (EAS formulation) and assumed natural strains (ANS formulation) to alleviate the different types of locking phenomena in solid-shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by incorporating an assumed stress field in a standard EAS formulation based on the FHW principle. To assess the present formulation’s accuracy, a variety of popular numerical benchmark examples related to element patch tests, convergence, mesh distortion, shell and laminated composite analyses are investigated and the results are compared with those available in the literature. This assessment reveals that the proposed solid-shell formulation provides very accurate results for a wide range of structural analyses.