Hybrid Stress Element Research Articles

A NEW FORMULATION OF ISOPARAMETRIC FINITE ELEMENTS AND THE RELATIONSHIP BETWEEN HYBRID STRESS ELEMENT AND INCOMPATIBLE ELEMENT

A new method of formulating isoparametric finite element is developed, and the element strains are proposed to be resolved into two parts, constant part and higher-order one. The new method indicates two important properties of isoparametric finite element, and the equivalent relationship between hybrid stress elements and incompatible elements. © 1997 by John Wiley & Sons, Ltd.

Admissible matrix formulation — from orthogonal approach to explicit hybrid stabilization

Admissible matrix formulation is a patch test approach for efficient construction of multi-field finite element models. In hybrid stress and strain elements, the formulation employs the patch test to identify the constraints on, respectively, the flexibility and stiffness matrices which are most detrimental to the element efficiency. Admissible changes are introduced to the matrices so as to reduce the computational cost while the element accuracy remains virtually intact. In this paper, a comprehensive review of admissible matrix formulation is presented. Finite element techniques seminal to the formulation are also discussed.

Penalty-equilibrating approach and an innovative formulation of 4-noded hybrid stress elements

The paper presents a penalty-equilibrating approach which is used to enforce the equilibrium constraints to individual hybrid elements so as to optimize the element performance. The approach has been applied successfully to improve the 4-noded hybrid stress membrane element (P-S) by Pian and Sumihara. It turns out that some defects of the P-S element, e.g. the shear locking phenomenon and the inability to pass the MacNeal's bending test, can be overcome and the element performance is significantly improved, without having to add any extra variables to the P-S element or to modify its stress fields.

Penalty‐equilibrating approach and an innovative formulation of 4-noded hybrid stress elements

The paper presents a penalty-equilibrating approach which is used to enforce the equilibrium constraints to individual hybrid elements so as to optimize the element performance. The approach has been applied successfully to improve the 4-noded hybrid stress membrane element (P-S) by Pian and Sumihara. It turns out that some defects of the P-S element, e.g. the shear locking phenomenon and the inability to pass the MacNeal's bending test, can be overcome and the element performance is significantly improved, without having to add any extra variables to the P-S element or to modify its stress fields.

Free vibration analysis of patched cracked composite laminates using a multilayer hybrid-stress finite element method

An assumed hybrid-stress finite element model incorporating two types of modified composite multilayer elements (MLTUP and MLTPH) is developed to investigate the free vibration of doubly and singly patched cracked laminates. Three-dimensional stresses of the laminated panel, patch and adhesive are accounted for. A composite patch material which has an efficient patching characteristic is employed. The effect of the patch parameters (length, thickness, width and position) on the fundamental natural frequency is also investigated. For a simply supported condition, the variation of the fundamental natural frequency after patching is negligible. However, the maximum variation of the fundamental natural frequency after patching for the problems solved under the cantilever edge condition is ca 10%. When the patch parameters are selected appropriately, the fundamental natural frequency after patching may become the same as that of the original uncracked laminate. This illustrates that the composite patch studied is a good choice for patching the cracked laminate tackled. The variations of the fundamental natural frequency for various crack lengths after patching are additionally presented. The technique developed can further be employed to treat a more realistic patching problem owing to that based on a three-dimensional viewpoint.

Multilayer hybrid-stress finite element analysis of composite laminates with delamination cracks originating from transverse cracking

An assumed hybrid-stress finite element model together with a composite multilayer element incorporating the technique of contact analysis is employed to analyze a composite laminate with delamination cracks originating from transverse cracking. Contact without friction on the crack surfaces and three-dimensional interlaminar stresses are taken into account. Once the contact zone exists, the assumed stress field satisfies the traction reciprocity conditions on the contact area and a transformation matrix is employed to deal with the contact conditions for the displacements on the contact zone. The interlaminar stresses and stress intensity factors are calculated accurately. As examples, cross-ply composite laminates with delamination cracks originating from transverse cracking under extension are studied. The influence of ply thickness on the growth of delamination cracks is also evaluated. The comparisons between the present results and referenced solutions show the high accuracy and validity of the present technique.

On optimization approaches of hybrid stress elements : Chang-Chun Wu & Y. K. Cheung, Finite Elements in Analysis & Design, 21(1–2), 1995, pp 111–128

On optimization approaches of hybrid stress elements : Chang-Chun Wu & Y. K. Cheung, Finite Elements in Analysis & Design, 21(1–2), 1995, pp 111–128

Filament-wound composite links in multibody systems

Dynamics of multi-link systems consisting of filament-wound composite beams and rigid bodies undergoing large rotations and translations is formulated. A two-node hybrid-stress shear-flexible finite element is used in modeling the links. The mass properties of the finite element which are required for multibody dynamic analysis are developed. A recursive formulation based on Kane's equations is used in the analysis of the multi-link system. Closed loops and prescribed motions are allowed in the system and the corresponding constraint equations are imposed. The constraint violations due to numerical errors are stabilized by using Baumgarte's method. A slider-crank mechanism is simulated to illustrate the differences between the responses of isotropic and composite mechanisms of the same weight.

A displacement scheme with drilling degrees of freedom for plane elements

AbstractA scheme of boundary displacements with drilling degrees of freedom for plane elements is presented. The scheme is free from zero displacement modes and allows the development of hybrid finite elements with vertex and mid‐side nodes, each node including a drilling degree of freedom besides the translational ones. Four quadrilateral isoparametric hybrid stress elements are implemented, and numerical results for some current test problems are given.

On optimization approaches of hybrid stress elements

A rational energy constraint to be used as the optimization condition (OPC) for hybrid finite element is presented. Based on OPC, the optimal stress pattern of the element is established in a general formulation. In this approach, the improvement of the element performance is carry out by a pretreatment of the initially assumed stress field. As an alternative way for the optimization of hybrid elements, the penalty-equilibrating approach is also suggested in the paper in which the equilibrium equation is enforced to the individual elements directly. Finally, a discussion on the stress distribution in an element is made. Some requirements on the rational σ-distribution are suggested. All of the points mentioned have been verified by means of some innovative hybrid elements in 2-D, 3-D and axisymmetric problems.

A hybrid-stress finite element for nonuniform filament-wound composite box-beams

A two-node, hybrid-stress finite element is formulated for static and dynamic analyses of linearly tapered, filament-wound composite box-beams. The effects of shear deformation and rotatory inertia are taken into account. The extensional, flexural and torsional actions are modeled. The element is assessed by a number of static and dynamic test problems for uniform and nonuniform, isotopic and composite box-beams. The results show the accuracy of the formulation.

Stress analyses of solids with rectangular holes by 3-D special hybrid stress elements

Two kinds of special 3-dimensional 12-node finite elements-each one contains a traction-free planar surface-have been developed based on Hellinger-Reissner principle by assumed stress hybrid method. Example solutions have demonstrated the advantage of using these special elements for analyzing plates and solids with rectangular holes.

New axisymmetric solid elements based on an extended Hellinger-Reissner principle

4-node incompatible axisymmetric hybrid stress elements are developed by a modified rational approach based on an extended Hellinger-Reissner principle. The stress fields are derived by using the constraint conditions that the equilibrium equations of higher order stress terms are imposed in a variational sense through the additional incompatible displacements. The resulting elements enjoy good performance in bending problems and at the nearly incompressible limit. Several examples are given to demonstrate the performance of the new elements.

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

AbstractAn 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.

On alternative hybrid stress 2d and 3d elements

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.

Hybrid stress formulation for higher-order theory of laminated shell analysis

Based upon a modified Hellinger-Reissner principle, a new four-node hybrid stress element is proposed for the linear elastic analysis of laminated orthotropic or anisotropic plates and shells. A constrained initial stress trial is introduced so that equilibrium constraints can be relaxed by the variational principle. Thus the complicated work of selecting an equilibrating stress field can be avoided. The example problems illustrate the performance of the element which is seen to be reliable and efficient.

Hybrid Plane Quadrilateral Element with Corner Rotations

A quadrilateral plane element for analysis of spatial structures is often formulated by combining a plane stress (membrane) element with two translational inplane degrees of freedom at each corner to a plate bending element with two rotational degrees of freedom and one translational (normal to the plane of element) at each corner. An additional rotational degree of freedom normal to the plane of the element and a fictitious torsional spring are added at each corner. This brings the total corner nodal degrees of freedom to three rotations and three translations and makes the element usable for general spatial structures. A quadrilateral hybrid stress element with two translational and one rotational degrees of freedom at each corner is presented in this paper. When used as the membrane constituent of a shell element, the rotational degree of freedom provides factual torsional stiffness. The element matrices are derived by minimization of a functional consisting of displacement and stress fields. Allman's displacement field is used with seven stress modes. The element passes the patch test and is invariant to orientation of axes and node numbering. Results of commonly employed benchmark tests are presented to show that the element is accurate and robust.

Analysis of laminated composite plates by hybrid stress isoparametric element

A simple quadrilateral hybrid stress element is presented for the analysis of laminated orthotropic and anisotropic plates. The initial trials of layer stresses are pretreated by an energy constraint. A higher-order theory incorporating realistic through thickness approximation of the inplane displacement is proposed. A variety of laminated plate problems are solved and the results are compared with the analytical solution, which demonstrate the validity of the method.

Rectangular hybrid shell element for analysing folded plate structures

A new rectangular hybrid stress element for analysing folded plate structures is presented. The new four-node element, which is based on the classical hybrid stress method, is called the hybrid coupling element and is generated by a combination of a hybrid plane stress element and a hybrid plate bending element. An invariant equilibrated stress field in each element and an inter-element compatible boundary displacement field are assumed independently in this model. Two additional degrees of freedom at the two nodes which lie along the line of intersection of two planes enable the displacement compatibility. They correspond to the rotation about the local z-axis of the element and are called ‘drilling’ degrees of freedom. The new hybrid finite element system has been used for an investigation of the response of a variety of elastic mountings and has yielded reliable results compared with classical methods. Finally, the advantages of such mountings in engineering design are emphasized.

A TWO‐FIELD SOLID ELEMENT SUITING THIN‐MESH ANALYSIS BY ADMISSIBLE MATRIX FORMULATION

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.