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.     

Plane isoparametric hybrid-stress elements: Invariance and optimal sampling

Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.     

Elastic-plastic analysis of plates by the hybrid-stress model and initial-stress approach

Two alternative hybrid-stress-based functionals are examined for the incremental elastic-plastic static analysis of single layer plates. Material nonlinear effects are incorporated via the initial-stress approach so that an equivalent nodal force vector is defined and the stiffness remains constant throughout the incremental loading. The alternative functionals differ in the incremental stress which is assumed to satisfy equilibrium; in the first, it is the actual stress increment, and in the second it is the elastic stress increment. Results are presented for two example problems, and comparisons of the alternative functionals and plausible iteration schemes are given. The effects of variation of pertinent solution parameters are also shown. A 4-node hybrid-stress plate element based on a Mindlin-type displacement field is used for most cases; however, limited results are also presented using an 8-node plate element, thus permitting comparisons of the relative efficiencies of the two elements.     

Three-dimensional hybrid-stress isoparametric quadratic displacement elements

Two 20-node quadratic displacement three-dimensional isoparametric elements are developed based on the hybrid-stress model. The elements differ in the stress interpolation used. In one case, the stress polynomials are selected to correspond approximately to the strain polynomials obtained from the displacement field, and a 57β stress field results. In the other element, complete cubic polynomials are used which are forced to satisfy the equilibrium equations and stress compatibility equations, and a 69β stress field results. Both elements possess correct rank, but only the 69β element is invariant. Results obtained using these two elements, and the corresponding 20-node assumed-displacement element, are compared and the 69β element is shown to be the better element. The 2 × 2 × 2 Gauss stations are also verified to be the optimal sampling points for these elements.     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

A C0 triangular plate element with one-point quadrature

A triangular plate element of the Mindlin type, which uses linear fields for rotations and transverse deflections, is developed. Success is achieved, in contrast to previous failures with this element, by decomposing the displacements into well-defined bending and shear modes which are associated with bending and shear energies, respectively. The element only requires a single quadrature point, which is very attractive for nonlinear analysis. Results are presented for a variety of plate problems; the accuracy is similar to the 4-node bilinear Mindlin plate and its rate of convergence is approximately of order h².     

Improved hybrid-stress axisymmetric elements includng behaviour for nearly incompressible materials

A series of 4-node axisymmetric solid-of-revolution elements with quadrilateral cross-section are developed based on the hybrid-stress finite element model. The displacement interpolation is identical, and the elements differ in the stress field chosen; alternative schemes for selection of stress distribution are investigated. The performance of these elements is assessed and compared for the example problems of a thick cylinder and thick sphere under internal pressure. Of particular importance are convergence, intra-element stress distributions, and element performance as the incompressible state is approached.     

Eigenvalues and stable time steps for the bilinear Mindlin plate element

Eigenvalues are obtained for the 4-node Mindlin plate element with one-point quadrature. Both isotropic and some anisotropic matenals are considered, but for the latter only bounds are obtained on the eigenvalues. These eigenvalues provide stable time steps for explicit time integration algorithms.     

A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates

A new smoothed finite element method (S-FEM) is proposed using hybrid smoothing operations based on nodes and edges of the mesh for static and free vibration analyses of plates governed by the Reissner–Mindlin plate theory. In the present approach, both the node-based smoothed finite element method (NS-FEM) and edge-based smoothed finite element method (ES-FEM) are utilized in a careful designed manner to overcome the shear locking. The formulations use 3-node triangular elements for easy automatic mesh creation, and linear interpolation functions are used for simplicity and robustness. The bending strains field are smoothed by the means of gradient smoothing technique over smoothing domains constructed by element edges, while the shear strains filed is smoothed based on the combination of NS-FEM and ES-FEM with a proper weightage controlled by a coefficient. A simple formula is developed for automatic selection of the coefficient by considering mesh size and thickness of the plate. For easy reference, the present technique is termed as NS+ES-FEM. The numerical examples demonstrate that the proposed method passes the shear-locking test and improves accuracy of the solution.     

Hybrid stress reduced-Mindlin elements for thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non-normal cross-section rotations; as a result, simple C^o interpolation of displacement and rotation can be used, and the number of degrees-of-freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness-formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8-node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non-locking in the thin plate limit.     

Mindlin板弯曲和振动分析的高阶杂交应力三角形单元

现有的Mindlin板单元只能通过零剪力分片检验,而不能通过非零常剪力分片检验。该文根据Reissner- Mindlin一阶剪切变形理论,基于余能原理,提出了一种高阶杂交应力六节点三角形Mindlin板单元。该单元特点是不仅能通过零剪力分片检验,而且能通过严格的非零常剪力增强型分片检验。构造单元时特别注意了单元边界位移以及域内应力的插值函数的选取。采用任意阶Timoshenko梁函数作为边界位移插值函数,应力插值函数选取为满足平衡方程的多项式。对不同厚度不同边界条件的方板进行弯曲和自由振动分析,质量矩阵采用集中质量阵。数值结果表明无论对薄板还是中厚板,该单元均是准确有效的。     

A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates. The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements. The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation. A so-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed. Several numerical examples are used to illustrate the accuracy and effectiveness of the present method.     

A new hybrid-mixed variational approach for Reissner–Mindlin plates. The MiSP model

A valuable variational approach for plate problems based on the Reissner–Mindlin theory is presented. The new MiSP (Mixed Shear Projected) approach is based on the Hellinger–Reissner variational principle, with a particular representation of transversal shear forces and transversal shear strains. The approximations of the shear forces are derived from those of the bending moments using the corresponding equilibrium relations. The shear strains are defined in terms of the edge tangential strains that are projected on the element degrees of freedom. Two finite elements are developed on the MiSP approach basis: 3-node triangular element MiSP3 and 4-node quadrilateral element MiSP4. Both elements can be considered as the most simple among the existent mixed elements. A modified MiSP model with a derived 4-node element is also presented. Numerical experiments are presented which show that the MiSP elements do not exhibit shear locking and give excellent results for thick and thin plates. They also pass the patch test for a general triangle and quadrilateral. © 1998 John Wiley & Sons, Ltd.     

High order three-dimensional hybrid-stress elements for thick-plate analysis

Two alternative hybrid-stress based three-dimensional elements are presented. These elements are intended for use in the analysis of thick plates for which the through-thickness variation of displacement and stress may be of high order, and in which a single element is to be used through the thickness of the plate. For both elements, the displacement assumption allows for cubic through-thickness variation in the inplane displacements, and quadratic through-thickness variation for the transverse displacement. The elements differ in the stress assumption used. In the first element, the z variations in the stress components are chosen to be consistent with the z variations of the strain distribution calculated from the displacement assumption, and the traction boundary conditions at the upper and lower surfaces are satisfied in the weighted integral sense. In the second element, the z variations in the stress components are chosen to achieve consistency with respect to the equilibrium equations, and the traction-free conditions at the upper and lower surface of the plate are satisfied exactly. The example problem of a semi-infinite, simply-supported thick plate under transverse sinusoidal loading is considered. Results obtained by using each of these elements are compared with available elasticity and approximate solutions, and an assessment of the relative merits and range of applicability of each element is made.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

半圆孔板承受拉伸和弯曲时的三维应力集中

根据Hellinger-Reissner原理建立了具有一个无外力圆柱表面三维8节点杂交应力元,元内假定应力场满足以柱坐标表示的三维平衡方程及无外力圆柱面上的外力边界条件,当元退化为二维时也满足协调方程。单元位移场选取与相邻元协调。用这种特殊杂交应力元,在相当粗的网格下即能准确地分析具有半圆孔厚(薄)板的三维(二维)应力集中。     

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allmans membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.     

Elastic-plastic analysis of stress state and spread of plastic zone around single edge-cracked plate by finite element method

A two-dimensional finite element model for estimating elastic-plastic displacements and stresses near a crack for the elastic-plastic situation of loadings is presented. Singularity effects near the crack tip are solved by introducing 12-node cubic isoparametric elements in the crack tip region and collapsing them into triangular elements. The proposed finite element model is used to determine the spread of the plastic zone and mouth opening displacements of an edge-cracked structural steel plate. The spread of the plastic zone is obtained by successive increments of applied nominal tensile stress transverse to the crack length. The mouth opening displacement values obtained by this model are also compared with those measured experimentally.     

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.     

Modified discrete Mindlin hypothesises for laminated composite structures

The present work deals with a procedure of defining discrete kinematic and mechanic hypothesises for Reissner-Mindlin plate finite elements. They are associated with the transverse shear strains for which the corresponding classical approximations caused the shear locking problem. Used in the past for isotropic homogeneous bending-shear plates, these hypothesises have been locally modified in order to be used for laminated composite plates. The derived plate finite element model, labelled DDM “Displacement Discrete Mindlin” has been applied to a 3-node triangular (DMTP: Discrete Mindlin Triangle for Plates) and validated on some standard problem tests, including one or several orthotropic layers.