A finite element model for Mindlin plates

In the general framework of Reissner-Mindlin theory, a plate model based on certain potential functions is discussed, together with its mechanical interpretation. A finite element implementation is also described and numerical results are reported.     

24-DOF quadrilateral hybrid stress element for couple stress theory

A 24-DOF quadrilateral hybrid stress element for couple stress theory is proposed in this study. In order to satisfy the equilibrium equation in the domain of the element, the $21\beta $ Airy stress functions are chosen a assumed stress interpolation functions, and beam functions are adopted as the displacement interpolation functions on the boundary. This element can satisfy weak C $^{0}$ continuity with second-order accuracy and weak $\hbox {C}^1$ continuity simultaneously. So the element can pass the enhanced patch test of a convergence condition. Moreover, the reduced integration and a stresses smooth technique are introduced to improve the element accuracy. Numerical examples presented show that the proposed model can pass the $\hbox {C}^{0-1}$ enhanced patch test and indeed possesses higher accuracy. Besides, it does not exhibit extra zero energy modes and can capture the scale effects of microstructure.     

A simple hybrid stress element for shear deformable plates

A new quadrilateral 4‐node element for shear deformable plates is developed based on the hybrid stress formulation. The element is designed to be simple, stable, free of locking and to pass all the patch tests. To this purpose, special attention is devoted to select displacement and stress approximations. The standard displacement interpolation is enhanced by linking the transverse displacement to the nodal rotations and an appropriate stress approximation is rationally derived. In particular, the assumed stress approximation is equilibrated within each element, co‐ordinate invariant and ruled by the minimum number of parameters. Excellent element performance is demonstrated by a wide experimental evaluation. Copyright © 2005 John Wiley & Sons, Ltd.     

Extension to linear dynamics for hybrid stress finite element formulation based on additional displacements

 Based upon legitimate variational principles, one microscopic-macroscopic finite element formulation for linear dynamics is presented by Hybrid Stress Finite Element Method. The microscopic application of Geometric Perturbation introduced by Pian and the introduction of infinitesimal limit core element (Baby Element) have been consistently combined according to the flexible and inherent interpretation of the legitimate variational principles initially originated by Pian and Tong. The conceptual development based upon Hybrid Finite Element Method is extended to linear dynamics with the introduction of physically meaningful higher modes.     

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     

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 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 stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid finite element model for multilayered plates

We present a finite element model for multilayered plates, based on a primal-hybrid variational formulation. Namely, each layer is analyzed as it were a lonely structure, and the displacement continuity is imposed from one layer to the other by means of Lagrange multipliers. Then, a Mindlin-like displacement field is assumed for any layer; the resulting continuous problem is proven to be well-posed under rather general hypotheses. Finally, a finite element model is deduced, using a very simple scheme (piecewise linear approximation for the displacement components and piecewise constant Lagrange multipliers). The numerical results assess the good performance of the proposed model.     

A new 12 DOF quadrilateral element for analysis of thick and thin plates

This paper presents a simple quadrilateral 12 DOF plate bending element based on a modified version of the hybrid-Trefftz approach. This element makes use of two independent fields of generalized displacements:

• i a non-conforming field (11 Trefftz terms for transverse displacement w and the corresponding rotations Θ_x, Θ_y) satisfying the governing differential equations of Reissner-Mindlin theory;
• ii an auxiliary conforming field with displacements w? linked to rotations $ \tilde \Theta _x,\tilde \Theta _y$, by the requirement of constant boundary distribution of the corresponding tangential component S?_t, of the transverse shear. This allows quadratic w? and linear $ \tilde \Theta _x,\tilde \Theta _y $, at the element boundary to be obtained with only 3 DOF at the corner nodes.

The resulting element, denoted by Q?21–11, is robust and free of shear locking in the thin limit. The numerical assessment involves comparison with several recently presented 12 DOF thick plate quadrilaterals as well as with the standard 16 DOF hybrid-Trefftz quadrilateral, Q21-15S, with 15 Trefftz terms and independently interpolated w? and $ \tilde \Theta _x,\tilde \Theta _y $.     

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

A new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case. Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.     

Partial hybrid stress element for the analysis of thick laminated composite plates

A new element—a partial hybrid stress element—is proposed in this paper for the analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger–Reissner principle through dividing six stress components into a flexural part (σ_x, σ_y, σ_xy, σ_z) and a transverse shear part (τ_xy, τ_yz). The element stiffness matrix can be formulated by assuming a stress field only for transverse shear stresses, while all the others are obtained from an assumed displacement field. Consequently, this new element combines the benefits of the conventional displacement method and the hybrid stress method. A twenty-node hexahedron element is employed in each layer for the displacement field. For the assumed transverse shear stress field, only the traction-free boundary conditions and interface traction continuity are satisfied. The equilibrium equation is enforced by the variational principle. Hence, the complicated work of searching an equilibrating stress field for all the six stress components in the hybrid stress method can be avoided. Furthermore, the interlaminar traction discontinuity, especially transverse shear, encountered by the conventional displacement method and higher-order plate element for laminated plate analysis can also be overcome. Examples are illustrated to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element.     

A simple and accurate four-node quadrilateral element using stabilized nodal integration for laminated plates

This paper reports the development of a simple but efficient and accurate four-node quadrilateral element for models of laminated, anisotropic plate behaviour within the framework of the first-order shear deformation theory. The approach incorporates the strain smoothing method for mesh-free conforming nodal integration into the conventional finite element techniques. The membrane-bending part of the element stiffness matrix is calculated by the line integral on the boundaries of the smoothing elements while the shear part is performed using an independent interpolation field in the natural co-ordinate system. Numerical results show that the element offered here is locking-free for extremely thin laminates, reliable and accurate, and easy to implement. Its convergence properties are insensitive to mesh distortion, thickness-span ratios, lay-up sequence and degree of anisotropy.     

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 incompatible and unsymmetric four-node quadrilateral plane element with high numerical performance

Recent studies show that the unsymmetric finite element method exhibits excellent performance when the discretized meshes are severely distorted. In this article, a new unsymmetric 4-noded quadrilateral plane element is presented using both incompatible test functions and trial functions. Five internal nodes, one at the elemental central and four at the middle sides, are added to ensure the quadratic completeness of the elemental displacement field. Thereafter, the total nine nodes are applied to form the shape functions of trial function, and the Lagrange interpolation functions are adopted as the incompatible test shape functions of the internal nodes. The incompatible test displacements are then revised to satisfy the patch test. Numerical tests show that the present element can provide very good numerical accuracy with badly distorted meshes. Unlike the existing unsymmetric four-node plane elements in which the analytical stress fields are employed, the present element can be extended to boundary value problems of any differential equations with no difficulties.     

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

C^0?1 patch test (Int. J. Numer. Meth. Engng 2004; 61 :433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C^0?1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C^0?1 patch test, the element displacement functions used to calculate strains must satisfy C⁰ continuity (or weak C⁰ continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C¹ continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C⁰ continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C^0?1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.     

A robust non‐linear mixed hybrid quadrilateral shell element

In the paper a non‐linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid‐surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in‐plane and bending patch test. Using Newton's method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non‐linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post‐buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A general degree hybrid equilibrium finite element for Kirchhoff plates

We present a family of hybrid equilibrium finite elements for the quasistatic linear elastic analysis of plates governed by Kirchhoff theory. The moments are approximated by self‐balanced polynomial fields of general degree, and in order to impose strong codiffusivity, the normal boundary rotations are approximated with complete polynomials of the same degree, whereas the transverse deflections use polynomials one degree lower. Furthermore, it is also necessary to include an independent approximation of the vertex translations. We show that the triangular form of this element is stable, that is, free from spurious kinematic modes, and the formulation that we present allows these elements to be used as a standard displacement element. Examples of computed values and convergence of the solutions are presented, which demonstrate the performance of these elements. Copyright © 2013 John Wiley & Sons, Ltd.