Assumed Stress Hybrid Finite Element Research Articles

Analysis of sandwich plates: A three-dimensional assumed stress hybrid finite element

An eight-node assumed stress solid element with rotational degrees of freedom is employed for analyses of sandwich plates consisting of stiff face and a comparatively flexible core material with the aim to accurately and efficiently capture stresses. The element formulation is based directly on an eight-node element. This direct formulation requires fewer computations than a similar element that is derived from an internal 20-node element in which the midside degrees of freedom are eliminated by expressing them in terms of displacements and rotations at corner nodes. The formulation is based on Hellinger-Reissner variational principle. Numerical examples are presented to show the validity and efficiency of the present element.

Hybrid Stress Finite Element Methods for Plate and Shell Structures

This paper is a survey of the use of assumed stress hybrid finite element methods for analyzing plate and shell structures. It describes the construction of Kirchhoff plate element by complementary energy principle, and of Reissner-Mindlin plate element by Hellinger-Reissner principle. Special hybrid stress elements are used to model joining of elements which are not coplanar. Such elements include the so-called semiLoof plate elements and 3D-solid elements tailored for analyzing thin plates and shells.

Some notes on the early history of hybrid stress finite element method

The first remark of this paper is to disclose a historical fact that the assumed stress hybrid finite element method pioneered by Pian [1] in 1964 was actually developed originally based on the Hellinger–Reissner principle and not on the complementary energy principle as indicated in the published paper. The other remark is that Dick Gallagher was the first person to prove the fact that the rectangular plane stress elements by Pian is identical to the Turner element presented in 1956. Finally, through a formulation by Hellinger–Reissner principle a formal proof is given of the fact that the Pian element is also identical to Wilson's incompatible element. Copyright © 2000 John Willey & Sons, Ltd.

Folded plate sandwich panel roof integrating passive solar glazings

It is possible to enhance the economics of passive solar glazings by integrating it with the roofing system, thereby reducing the duplication of material and construction time. In this study, a folded plate sandwich panel roof has been chosen to integrate the passive solar glazings at an appropriate angle. The various factors involved in the orientation of glazings are discussed. The objective of this study is to develop the analysis of a roof structure with openings to suitably integrate the glazings. An analytical roof model is chosen and a stress analysis is performed to study the load deformation characteristics of the structural system. An assumed stress hybrid finite element program has been used for the analysis. The effect of various material properties on the structural behaviour of the system has been studied and the results are presented.

Rectangular hybrid elements for the analysis of sandwich plate structures

The structural behaviour of sandwich plate structures are characterized by transverse shear deformations in the core. The assumed stress hybrid finite element technique is particularly suitable for developing sandwich plate bending elements. In the present study, rectangular three-layer sandwich plate elements have been formulated using simple assumed stress functions. Numerical test problems have been solved to examine the convergence property and suitability of these elements. The results are compared with that of a complete quadratic stress mode element and with analytical solutions. Six degrees of freedom per node shell elements are formulated by combining the plate bending elements with membrane elements. A folded plate sandwich panel roof has been analyzed using these elements and the results are compared with the experimental values. The use of simple stress function gives satisfactory results and reduces the size of the matrices to be used, the length of the program, and the computation time for the formulation of element stiffness matrices. Key words: sandwich panel, structural analysis, finite element method, stress hybrid approach, folded plates.

An equilibrium method for prediction of transverse shear stresses in a thick laminated plate

First two equations of equilibrium are utilized to compute the transverse shear stress variation through thickness of a thick laminated plate after in-plane stresses have been computed using an assumed quadratic displacement triangular element based on transverse inextensibility and layerwise constant shear angle theory (LCST). Centroid of the triangle is the point of exceptional accuracy for transverse shear stresses. Numerical results indicate close agreement with elasticity theory. An interesting comparison between the present theory and that based on assumed stress hybrid finite element approach suggests that the latter does not satisfy the condition of free normal traction at the edge. Comparison with numerical results obtained by using constant shear angle theory suggests that LCST is close to the elasticity solution while the CST is closer to classical (CLT) solution. It is also demonstrated that the reduced integration gives faster convergence when the present theory is applied to a thin plate.

Damage development in bolt bearing of composite laminates

The buildup of damage in bolt-loaded specimens has been investigated. Destructive evaluation by staining, dissection, and microscopy was used to evaluate the complete three-dimensional geometry of the damage zone at several load levels and loading conditions for glass and graphite fiber-reinforced epoxy laminates with the stacking sequence [ [O/ + 457 45] 5/0] s. The assumed stress hybrid finite element method was used to determine the elastic stress distributions, including simulations of no friction and no sliding. Three sets of tests are included: a fastened bolt joint in tension, a fastened bolt joint in compression, and a fastened bolt hole in tension. Complete damage characteristics and stress distributions are even ply by ply through the thickness as a function of load. The local strength around the hole boundary is examined pointwise by the Tsai-Hill criterion; the maximum stress criterion is then used to determine the failure mode. First-ply failure criteria are applied and compared with experiments. The three sets of tests indicate that the damage zones of the net-tension and bearing-compression regions are independent at both the initiation and failure stages. Implications for gross failure criteria are discussed, and the need for more sophisticated methods is demonstrated. Attempts have been made to correlate actual damage zones with Whitney-Nuismer distance parameters, but no physical interpretation of these parameters appears justified.

Bearing/contact for anisotropic materials

The bearing/contact behavior of fiber composites has been investigated using a modified nonlinear assumed stress hybrid finite element formulation. The problem considered is an anisotropic body subjected to compressive contact from an isotropic indentor, with moderate sliding on the contact surface. The characteristic matrices of the bodies brought into contact are obtained by employing a modified principle of the minimum complementary energy, and using the contact conditions along the contacting boundary as constraints with corresponding Lagrangian multipliers. A computer program has been developed for the two-dimensional punch problem. The stick-slide behavior and frictional effects are performed using an iterative technique. Thirty cases of the punch/contact problem have been analyzed. Results for typical variations in properties show that, for the same frictional condition, the material stiffness in the indentation direction dominates the stresses on the contacting surface. By increasing the friction on the contact surface, the transverse tensile stress is relaxed; this can be directly related to suppression of fiber-resin debonding or splitting in practical bearing problems.

Mixed finite element models for plate bending analysis theory

The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.

Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method

Two assumed stress hybrid finite element models for analyzing the large deflection, linear elastic, static behavior of structures have been developed: a consistent model which satisfies the entire stress equilibrium equation and an inconsistent model which satisfies only the linear portion of this equation. These models are derived for two separate coordinate frames: a stationary system and a convected, updated system. Throughout the development “correction” terms are maintained in all the functionals to minimize the drifting of the approximate solution from the true solution. These correction terms correspond to checks on the stress equilibrium and compatibility in the reference state. Utilizing a tangent stiffness approach various incremental and incremental-iterative solution procedures are used. The actual applications utilize flat and shallow elements to analyze the large deflection (moderate rotation), small strain behavior of thin, linearly elastic beams and shells. For the beam problems a shallow curved, two node, six degree of freedom element is used. For the shell analysis flat and shallow shell elements are used. They are three node, fifteen degree of freedom, triangular elements. Example studies include comparisons of the consistent and inconsistent models, the flat and shallow elements, the two coordinate systems, the effectiveness of the correction terms and solution procedures, and the adequacy of the models and methods. The results demonstrate that the consistent and inconsistent models yield essentially the same results. Overall, the models yield satisfactory results with simple elements.

Alternate Hybrid-Stress Elements for Analysis of Multilayer Composite Plates

Two quadrilateral-shaped multilayer plate bending/stretching elements based on the assumed-stress hybrid finite-element model are presented. These elements may be applied to the analysis of fiber composite plates and shells composed of an arbitrary number of layers and arbitrary mate rial properties and fiber direction within each layer. Both elements incor porate the effects of transverse shear by including the transverse shear stresses τ xz and τyz, and by assuming independent rotations of the normals to the plate midsurface, so that normals to the midsurface in the un deformed state need not be normal to the midsurface after deformation. In the formulation of an element based on the assumed-stress hybrid model, the stress distribution in the interior of the element is expressed in terms of a finite number of stress parameters such that equilibrium is satisfied, and the displacement distribution on the boundary of the element is expressed in terms of generalized nodal displacements such that interelement displacement compatibility is maintained. The principle difference between the two elements presented is the level of approxima tion incorporated into the stress and displacement assumptions. For one element the stresses within each layer are related to a set of stress param eters within that layer, and the boundary displacement assumption allows for independent cross-sectional rotations within each layer (and is thus capable of modeling the severe cross-sectional warping often associated with thick laminated plates). For the other element, the stresses within each layer are related to a set of stress parameters associated with the entire laminate, and the boundarv displacement assumption allows for a uniform rotation of the cross section of the laminate (not necessarily normal to the plate midsurface). To assess the behavior of the two elements, two laminated composite plate example problems are considered, for which analytic solutions are available. The accuracy of predicted stress and displacement results obtained by using the two hybrid elements are compared with the exact solutions for cases involving both thick and moderately thick plates. Com parisons of the relative accuracy and computing efficiency of the two elements are presented, and the limitations on the use of each element are also discussed.

Analysis of developable shells with special reference to the finite element method and circular cylinders

The paper begins by noting that the practical and efficient numerical analysis of thin walled shells is far from a reality. Groundwork for the investigation starts with an examination of existing sufficiency conditions for convergence of the finite element method of analysis with refinement of mesh size; new and more practical conditions are then given specifically for shells. Working formulae of a suitable first approximation theory for the linear small deflexion behaviour are then given for arbitrary shells in lines of curvature and in geodesic coordinates. A variational principle is introduced which is more general than that for the well known assumed stress hybrid finite element model; its purpose is to provide a means to overcome the excessive rank deficiency which is sometimes encountered in the derive element stiffness matrix. , The formulae are next specialized to general developable shells for they are tne simplest to analyse and frequently occur in technology. Emphasis is given to the derivation of general formulae governing inextensional deformation, membrane action and rigid body movement because these constitute important factors in any adequate numerical analysis. . . , Specific application is made to circular cylindrical shells by first considering the interpolation of the kinematic continuity conditions along an arbitrary geodesic line. Details and numerical examples are provided for the first known fully compatible lines of curvature rectangular finite element which directly recovers arbitrary rigid body movements as well as inextensional deformations and membrane actions. The paper concludes with details and numerical examples of an arbitrarily shaped triangular finite element which employs the above mentioned variational principle m conjunction with linearly varying stress fields. All the rigid body movements are directly recovered as well as inextensional deformations and membrane actions. It is anticipated that this finite element and its derivatives will find widespread application.

An assumed stress hybrid finite element method for an incompressible and near-incompressible material

A variational principle based on assumed stress hybrid method suitable for incompressible or nearincompressible solid is formulated for finite element analysis. One example to illustrate the use of the method is given.