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A higher-order Reissner-Mindlin plate element method is presented based on the framework of assumed stress quasi-conforming method and Hellinger-Reissner variational principle. A novel six-node triangular plate element is proposed by utilizing this method for the static and free vibration analysis of Reissner-Mindlin plates.

First, the initial assumed stress field is derived by using the fundamental analytical solutions which satisfy all governing equations. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations after the strains are derived by using the constitutive equations. Finally, the arbitrary order Timoshenko beam function is adopted as the string-net functions along each side of the element for strain integration.

The proposed element can pass patch test and is free from shear locking and spurious zero energy modes. Numerical tests show that the element can give high-accurate solutions, good convergence and is a good competitor to other models.

This work gives new formulations to develop high-order Reissner-Mindlin plate element, and the new strategy exhibits advantages of both analytical and discrete methods.

A new laminated composite plate finite element is proposed that is numerically stable and accurate in displacements and stresses, including transverse shear stress. The formulation is based on the Hellinger—Reissner principle with Mindlin kinematics. All stress components are given independent approximations and do not satisfy equilibrium conditions a priori. A novel feature of the formulation is the additive decomposition of the displacement field into two parts corresponding to nodal interpolations and independent local basis functions. The additional basis functions and their associated parameters play an important role in characterizing the accuracy of the element. These functions eliminate shear locking in the lower order elements and provide additional variational constraints on the stresses, leading to very accurate results. A 4‐node and a 9‐node version are developed and it is shown that both elements pass the patch test suggested by Zienkiewicz et al. and are stable in the sense of the Babuška—Brezzi condition. The special structure of the element flexibility matrix provides computational efficiency approaching that of displacement based formulations.

The purpose of this paper is to present eight local elasto‐plastic beam element formulations incorporated into the corotational framework for two‐noded three‐dimensional beams. These formulations capture the warping torsional effects of open cross‐sections and are suitable for the analysis of the nonlinear buckling and post‐buckling of thin‐walled frames with generic cross‐sections. The paper highlights the similarities and discrepancies between the different local element formulations. The primary goal of this study is to compare all the local element formulations in terms of accuracy, efficiency and CPU‐running time.

The definition of the corotational framework for a two‐noded three‐dimensional beam element is presented, based upon the works of Battini .The definitions of the local element kinematics and displacements shape functions are developed based on both Timoshenko and Bernoulli assumptions, and considering low‐order as well as higher‐order terms in the second‐order approximation of the Green‐Lagrange strains. Element forces interpolations and generalized stress resultant vectors are then presented for both mixed‐based Timoshenko and Bernoulli formulations. Subsequently, the local internal force vector and tangent stiffness matrix are derived using the principle of virtual work for displacement‐based elements and the two‐field Hellinger‐Reissner assumed stress variational principle for mixed‐based formulations, respectively. A full comparison and assessment of the different local element models are performed by means of several numerical examples.

In this study, it is shown that the higher order elements are more accurate than the low‐order ones, and that the use of the higher order mixed‐based Bernoulli element seems to require the least number of FEs to accurately model the structural behavior, and therefore allows some reduction of the CPU time compared to the other converged solutions; where a larger number of elements are needed to efficiently discretize the structure.

The paper reports computation times for each model in order to assess their relative efficiency. The effect of the numbers of Gauss points along the element length and within the cross‐section are also investigated.

Based on the recent advances of hybrid stress finite elements, a series of alternative stress assumptions for these elements are investigated. Several new element models are proposed by using different concepts for the stress interpolation. Under a unified formulation presented in this paper for Hellinger—Reissner principle based hybrid stress element models, the element series 5β‐family for plane stress and 18β‐family for three‐dimensional problems are discussed. The extra incompatible displacements sometimes also added are not introduced in this unified formulation. A number of popular benchmark elastic problems are examined for both two element families. In each family, the element model presented in this paper using normalized transformed higher order stress trials usually gives better predictions than the others.

A hexahedral 8‐node element based on the Hellinger—Reissner principle is formulated with the γ projection operator so that it can achieve engineering accuracy for plate and beam problems with a single layer of elements. It passes the patch test and is less sensitive to mesh shape since the local coordinates are used to describe the stress fields. The resulting element stiffness is simple and only 3×3 submatrix inversions are needed. Numerical results show that the new element is both accurate and efficient.

To increase the use of the meshless method, a hybrid stress method is introduced into the meshless method.

The method is based on the radial point interpolation method (RPIM). According to the Hellinger Reissner principle, stress functions are introduced into the solution procedure. Finite elements are used as background cells for integration. All cells are divided into two types – the H cells, which are around the traction-free circular boundary, and the G cells. For the H cells, stress functions in polar coordinates are created. For the G cells, 12-parameter stress functions in Cartesian coordinates are used. Stress functions are based on equilibrium equations and stress compatible equation.

Numerical results show that this method is reliable.

Hybrid stress methods have been applied to finite element methods, but the finite element methods have not been applied into meshless methods.

Classical mixed formulations of the boundary‐value problem of linear elasticity are reviewed, and a new three‐field formulation is introduced. The formulation is an extension of the classical Hu‐Washizu approach, and takes the form of a non‐standard mixed problem. Convergence of finite element approximations of both the old and new methods are discussed, with an emphasis on their behaviour in the incompressible limit. Conditions for the stability and uniform convergence of the new method are presented, and it is shown that the Pian‐Sumihara basis, when used in the new formulation, leads to a convergent method.

A simple shape-free high-order hybrid displacement function element method is presented for precise bending analyses of Mindlin–Reissner plates. Three distortion-resistant and locking-free eight-node plate elements are proposed by utilizing this method.

This method is based on the principle of minimum complementary energy, in which the trial functions for resultant fields are derived from two displacement functions, F and f, and satisfy all governing equations. Meanwhile, the element boundary displacements are determined by the locking-free arbitrary order Timoshenko’s beam functions. Then, three locking-free eight-node, 24-DOF quadrilateral plate-bending elements are formulated: HDF-P8-23β for general cases, HDF-P8-SS1 for edge effects along soft simply supported (SS1) boundary and HDF-P8-FREE for edge effects along free boundary.

The proposed elements can pass all patch tests, exhibit excellent convergence and possess superior precision when compared to all other existing eight-node models, and can still provide good and stable results even when extremely coarse and distorted meshes are used. They can also effectively solve the edge effect by accurately capturing the peak value and the dramatical variations of resultants near the SS1 and free boundaries. The proposed eight-node models possess potential in engineering applications and can be easily integrated into commercial software.

This work presents a new scheme, which can take the advantages of both analytical and discrete methods, to develop high-order mesh distortion-resistant Mindlin–Reissner plate-bending elements.

A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996‐1999.

In this paper, a hybrid stress 12‐node brick element is presented. Its assumed stress field is derived by first examining the deformation modes of a geometrically regular element and then generalizing to other element configurations using tensorial transformation. The total number of stress modes is 30 which is minimal for securing the element rank. To reduce the computational cost associated with the fully populated flexibility matrix, the admissible matrix formation is employed to induce high sparsity in the matrix. Popular beam bending benchmark problems are examined. The proposed elements deliver encouraging accuracy.