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Gives 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. The range of applications of FEMs in this area is wide and cannot be presented in a single paper; therefore aims to give the reader an encyclopaedic view on the subject. The bibliography at the end of the paper contains 2,025 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 1992‐1995.

An 8‐node solid element applicable for thin structures is presented. The element employs eighteen assumed stress modes and the conventional displacement interpolation. The formulation starts with the hybrid stress element proposed by Pian and Tong. The higher order stress modes are first decomposed into the ones which do and do not lead to thin‐element locking. The recently established methodology of admissible matrix formulation allows the decoupling of the above two categories of stress modes in the flexibility matrix without triggering element instability or failure of the patch test. The element stiffness can thus be decomposed into a series of matrices. Locking can be eliminated by adjusting the magnitude of the pertinent matrices. Accuracy and convergence rate of the present element are found to be competent to many of the existing plate and shell models.

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.

This paper aims to describe the formulation of a displacement-based triangular membrane finite element with true drilling rotational degree of freedom (DOF).

The presented formulation incorporates the true drilling rotation provided by continuum mechanics into the displacement field by way of using the polynomial interpolation. Unlike the linked interpolation, that uses a geometric transformation between displacement and vertex rotations, in this work, the interpolation of the displacement field in terms of nodal drilling rotations is obtained following an unusual approach that does not imply any presumed geometric transformation.

New relationship linking the mid-side normal displacement to corner node drilling rotations is derived. The resulting new element with true drilling rotation is compatible and does not include any problem-dependent parameter that may influence the results. The spurious zero-energy mode is stabilized in a careful way that preserves the true drilling rotational degrees of freedom (DOFs).

Several works dealing with membrane elements with vertex rotational DOFs have been published with improved convergence rate, however, owing to the need for incorporating rotations in the finite element meshes involving solids, shells and beam elements, having finite elements with true drilling rotational DOFs is more appreciated.

The purpose of this paper is to extend a recent unsymmetric 8-node, 24-DOF hexahedral solid element US-ATFH8 with high distortion tolerance, which uses the analytical solutions of linear elasticity governing equations as the trial functions (analytical trial function) to geometrically nonlinear analysis.

Based on the assumption that these analytical trial functions can still properly work in each increment step during the nonlinear analysis, the present work concentrates on the construction of incremental nonlinear formulations of the unsymmetric element US-ATFH8 through two different ways: the general updated Lagrangian (UL) approach and the incremental co-rotational (CR) approach. The key innovation is how to update the stresses containing the linear analytical trial functions.

Several numerical examples for 3D structures show that both resulting nonlinear elements, US-ATFH8-UL and US-ATFH8-CR, perform very well, no matter whether regular or distorted coarse mesh is used, and exhibit much better performances than those conventional symmetric nonlinear solid elements.

The success of the extension of element US-ATFH8 to geometrically nonlinear analysis again shows the merits of the unsymmetric finite element method with analytical trial functions, although these functions are the analytical solutions of linear elasticity governing equations.

Poor bending response is a major shortcoming of lower-order elements due to excessive representation of shear stress/strain field. Advanced finite element (FE) formulations for classical elasticity enhance the bending response by either nullifying or filtering some of the symmetric shear stress/strain modes. Nevertheless, the stress/strain field in Cosserat elasticity is asymmetric; consequently any attempt to nullify or filter the anti-symmetric shear stress/strain modes may lead to failure in the constant couple-stress patch test where the anti-symmetric shear stress/strain field is linear. This paper aims at enhancing the bending response of lower-order elements for Cosserat elasticity problems.

A four-node quadrilateral and an eight-node hexahedron are formulated by hybrid-stress approach. The symmetric stress is assumed as those of Pian and Sumihara and Pian and Tong. The anti-symmetric stress components are first assumed to be completely linear in order to pass the constant couple-stress patch test. The linear modes are then constrained with respect to the prescribed body-couple via the equilibrium conditions.

Numerical tests show that the hybrid elements can strictly pass the constant couple-stress patch test and are markedly more accurate than the conventional elements as well as the incompatible elements for bending problems in Cosserat elasticity.

This paper proposes a hybrid FE formulation to improve the bending response of four-node quadrilateral and eight-node hexahedral elements for Cosserat elasticity problems without compromising the constant couple-stress patch test.

This paper aims to present an extension of two recently published elements (which are based on Petrov‐Galerkin formulation) to geometric nonlinear (GNL) problems.

Two different sets of shape functions, namely isoparametric and metric, suitably chosen to satisfy the necessary compatibility and completeness conditions, are used as test and trial functions, respectively. Total Lagrangian formulation is used for the implementation of the element.

In implementing the unsymmetric formulation for nonlinear problems, the deformation gradient tensor can be evaluated invariably using either isoparametric or metric shape functions. The developed elements are found to exhibit improved performance in the presence of mesh distortions.

The numerical problems in this paper involve linear elastic materials.

Extension of US‐QUAD8 and US‐HEXA20 for GNL problems is new.

Based on the orthogonal approach for hybrid element methods, refined three‐dimensional isoparametric hybrid hexahedral elements have been developed. The behaviour of the proposed models are discussed in respect of coordinate invariance, spurious zero energy modes and the ability to pass the patch test. By adopting the orthogonality of strain energy, the element stiffness matrix can be decomposed into a series of stiffness matrices in which the implementational effectiveness can be improved. A number of examples is used to demonstrate the implementation efficiency, accuracy and distortion insensitivity of the proposed elements, and its capacity of handling nearly incompressible materials

Numerical experiences reveal that the performances of the dynamic relaxation (DR) method are related to the structural types. This paper is devoted to compare the DR schemes for geometric nonlinear analysis of shells. To achieve this task, 12 famous approaches are briefly introduced. The differences among these schemes are between the estimation of the time step, the mass and the damping matrices. In this study, several benchmark structures are analyzed by using these 12 techniques. Based on the number of iterations and the analysis duration, their performances are graded. Numerical findings reveal the high efficiency of the kinetic DR (kdDR) approach and Underwood’s strategy.

Up to now, the performances of various DR algorithms for geometric nonlinear analysis of thin shells have not been investigated. In this paper, 12 famous DR methods have been used for solving these structures. It should be noted that the difference between these approaches is in the estimation of the fictitious parameters. The aforementioned techniques are used to solve several numerical samples. Then, the performances of all schemes are graded based on the number of iterations and the analysis duration.

The final ranking of each strategy will be obtained after studying all numerical examples. It is worth emphasizing that the number of iterations and that of convergence points of the arc length algorithms are dependent on the value of the initial arc length. In other words, a slight change in the magnitude of the arc length may lead to the wrong responses. Contrary to this behavior, the analyzer’s role in the dynamic relaxation techniques is considerably less than the arc length method. In the DR strategies when the answer approaches the limit points, the iteration number increases automatically. As a result, this algorithm can be used to analyze the structures with complex equilibrium paths.

Numerical experiences reveal that the DR method performances are related to the structural types. This paper is devoted to compare the DR schemes for geometric nonlinear analysis of shells.

Geometric nonlinear analysis of shells is a sophisticated procedure. Consequently, extensive research studies have been conducted to analyze the shells efficiently. The most important characteristic of these structures is their high resistance against pressure. This study demonstrates the performances of various DR methods in solving shell structures.

Up to now, the performances of various DR algorithms for geometric nonlinear analysis of thin shells are not 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.