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The mixed assumed strain approach proposed by Simo and Rifai is used to derive three 8‐noded hexahedral mixed strain elements. The approach is also generalized to geometrically non‐linear problems. Based on the Galerkin form of Hu‐Washizu three field variational principle, the Green‐Lagrange strain tensor and the second Piola‐Kirchhoff stress tensor (symmetric) are employed to develop the geometrically non‐linear formulation for 2D and 3D mixed enhanced strain elements. Numerical results are presented to show that the resulting hexahedral mixed strain elements possess all the ideal qualities. They are able to pass the patch test, do not exhibit the false shear phenomena and do not lock for nearly incompressible materials. Also, they are less sensitive to distorted meshes than standard isoparametric elements and exhibit high accuracy for both linear and non‐linear problems, permitting coarse discretizations to be utilized. The elements developed in this paper have been implemented in the general purpose FE package LUSAS.

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.

This paper aims to propose a new robust membrane finite element for the analysis of plane problems. The suggested element has triangular geometry. Four nodes and 11 degrees of freedom (DOF) are considered for the element. Each of the three vertex nodes has three DOF, two displacements and one drilling. The fourth node that is located inside the element has only two translational DOF.

The suggested formulation is based on the assumed strain method and satisfies both compatibility and equilibrium conditions within each element. This establishment results in higher insensitivity to the mesh distortion. Enforcement of the equilibrium condition to the assumed strain field leads to considerably high accuracy of the developed formulation.

To show the merits of the suggested plane element, its different properties, including insensitivity to mesh distortion, particularly under transverse shear forces, immunities to the various locking phenomena and convergence of the element are studied. The obtained results demonstrate the superiority of the suggested element compared with many of the available robust membrane elements.

According to the attained results, the proposed element performs better than the well-known displacement-based elements such as linear strain triangular element, Q4 and Q8 and even is comparable with robust modified membrane elements.

A critical assessment of the 4‐node assumed strain element as proposed by Dvorkin and Bathe is made. The element performed excellently in all investigated shell problems which sometimes caused difficulties for other assumed strain techniques. For efficient computation in the non‐linear range, linearization of the virtual work equation is done to yield the consistent tangent stiffness. The shell formulation is done for stress and strain tensors based on local element coordinates. To demonstrate the effectiveness and rapid convergence of the non‐linear formulation, three examples are tested for large displacements.

Implementation details of the assumed shear strain method in a novel finite rotation shell theory are discussed. Careful considerations of the pertinent aspects of the Newton solution procedure are given. The latter results in a very robust performance of the presented 4–node shell element in some challenging finite rotation problems.

The use of enhanced strains leads to an improved performance of low order finite elements. A modified Hu‐Washizu variational formulation with orthogonal stress and strain functions is considered. The use of orthogonal functions leads to a formulation with B (overline) ‐strain matrices which avoids numerical inversion of matrices. Depending on the choice of the stress and strain functions in Cartesian or natural element coordinates one can recover, for example, the hybrid stress element P‐S of Pian‐Sumihara or the Trefftz‐type element QE2 of Piltner and Taylor. With the mixed formulation discussed in this paper a simple extension of the high precision elements P‐S and QE2 to general non‐linear problems is possible, since the final computer implementation of the mixed element is very similar to the implementation of a displacement element. Instead of sparse B‐matrices, sparse B (overline) ‐matrices are used and the typical matrix inversions of hybrid and mixed methods can be avoided. The two most efficient four‐node B (overline) ‐elements for plane strain and plane stress in this study are denoted B (overline)(x, y)‐QE4 and B (overline)(ξ, η)‐QE4.

The purpose of this paper is to present the implementation in a finite element (FE) code of a recently developed material model for the analysis of cracked reinforced concrete (RC) panels. The model aims for the efficient nonlinear analysis of large‐scale structural elements that can be considered as an assembly of membrane elements, such as bridge girders, shear walls, transfer beams or containment structures.

In the proposed constitutive model, the equilibrium equations of the cracked membrane element are established directly at the cracks while the compatibility conditions are expressed in terms of spatially averaged strains. This allows the well‐known mechanical phenomena governing the behaviour of cracked concrete elements – such as aggregate interlock (including crack dilatancy effects), tensile fracture and bond shear stress transfer – to be taken into account in a transparent manner using detailed phenomenological models. The spatially averaged stress and strain fields are obtained as a by‐product of the local behaviour at the cracks and of the bond stress transfer mechanisms, allowing the crack spacing and crack widths to be obtained directly from first principles. The model is implemented in an FE code following a total formulation.

The fact that the updated stresses at the cracks are calculated explicitly from the current spatially averaged total strains and from the updated values of the state variables that are used to monitor damage evolution contributes to the robustness and efficiency of the implementation. Some application examples are presented illustrating the model capabilities and good estimates of the failure modes, failure loads, deformation capacity, cracking patterns and crack widths were achieved.

While being computationally efficient, the model describes the complex stress and strain fields developing in the membrane element, and retrieves useful information for the structural engineer, such as concrete and reinforcement failures as well as the crack spacing and crack widths.

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.

An efficient implementation of a constitutive model for reinforced concrete plates is discussed in this work. The constitutive model is set directly in terms of stress resultants and their energy conjugate strain measures, relating their total values. The latter simplification is justified by our primary goal being an evaluation of the limit load of a reinforced concrete plate. A concept of the ‘rotating crack model’ is utilized in proposing the constitutive model to relate the principal values of bending moments and the corresponding values of curvatures. Efficient implementation is provided by a very robust, but inexpensive plate element. The element is based on an assumed shear strain field and a set of incompatible bending modes, which provides that the non‐linear computations, pertinent to constitutive model, can be carried out locally, i.e. independently at each numerical integration point. Set of numerical examples is presented to demonstrate a very satisfying performance of the proposed model.

The purpose of this paper is to propose two linear solid-shell finite elements, a six-node prismatic element denoted SHB6-EXP and an eight-node hexahedral element denoted SHB8PS-EXP, for the three-dimensional modeling of thin structures in the context of explicit dynamic analysis.

These two linear solid-shell elements are formulated based on a purely three-dimensional (3D) approach, with displacements as the only degrees of freedom. To prevent various locking phenomena, a reduced-integration scheme is used along with the assumed-strain method. The resulting formulations are computationally efficient, as only a single layer of elements with an arbitrary number of through-thickness integration points is required to model 3D thin structures.

Via the VUEL user-element subroutines, the performance of these elements is assessed through a set of selective and representative dynamic elastoplastic benchmark tests, impact-type problems and deep drawing processes involving complex non-linear loading paths, anisotropic plasticity and double-sided contact. The obtained numerical results demonstrate good performance of the SHB-EXP elements in the modeling of 3D thin structures, with only a single element layer and few integration points in the thickness direction.

The extension of the SHB-EXP solid-shell formulations to large-strain anisotropic plasticity enlarges their application range to a wide variety of dynamic elastoplastic problems and sheet metal forming simulations. All simulation results reveal that the numerical strategy adopted in this paper can efficiently prevent the various locking phenomena that commonly occur in the 3D modeling of thin structural problems.