General Shell Element Research Articles

A new one-point quadrature, quadrilateral shell element with drilling degrees of freedom

In this paper we discuss the development of a new four-node general shell element with single point quadrature used for the analysis of non-linear geometrical and material problems. One of the main features of the present development is the implementation of a rotation component around the shell normal (i.e. drilling rotation) to Belytschko's family of shell elements. Thus, at each node, six degrees of freedom (i.e. three translations and three rotations) make the element easy to connect to space beams, stiffeners or intersecting shells. A projection scheme for warping correction is proposed so that the element is accurate for both flat and warped configurations. All locking phenomena (such as transverse shear, in-plane shear and membrane locking) are controlled by an assumed strain method. Also, a physical stabilization approach for control of spurious zero-energy modes is proposed without requiring any artificial stabilization parameters. This approach is inexpensive and accurate because it updates and stores hourglass stresses only at the mid-surface rather than at all the integration points through the shell thickness. To demonstrate the features of the new shell element, it was applied to the analysis of large-scale, non-linear, static, dynamic and impact problems using elastic, isotropic/anisotropic elastoplastic and composite damage models. The element performed well for the problems that sometimes cause difficulties for other shell element techniques.

Nonlinear analysis of shells using the MITC formulation

The formulation of general shell elements using the method of mixed interpolation of tensorial components (MITC) is reviewed. In particular three elements that were formulated using the MITC method are examined: the MITC4 and MITC8 that were developed for general nonlinear analysis under the restriction of small strains and the MITC4-TLH that was developed for finite strain elasto-plastic analysis of shells.

Local‐global analysis on localized non‐linear shells of revolution

AbstractA finite element program is developed as a tool to analyse shells of revolution with local non‐linearities. In reality, shells of revolution often exhibit local deviations, like a cut‐out, a junction and/or an imperfection. The stress concentration around a local deviation may produce plasticity and/or geometric non‐linearities in the surrounding region. The analytical model consists of three different types of elements: rotational, transitional and general. The rotational shell elements are used in the region where the shell is axisymmetrical and linear, while the two‐dimensional general shell elements are deployed in the deviation region where non‐linearities may occur. Transitional shell elements connect the two distinctively different types of elements to achieve displacement field continuities. The solution using the local‐global system with appropriate condensation and a predicted stress incremental procedure is suggested. It is shown that the technique is a very attractive alternative to the entirely general element style analysis for axisymetric shell structures with local deviations.

A local–global model for the non‐linear analysis of locally defective shells of revolution

AbstractIn this paper a local–global analysis technique is presented for the non‐linear analysis of shells of revolution with a localized material discontinuity in the form of a crack or a cutout. The local zone is modelled using two‐dimensional general shell elements. Axisymmetric shell elements with Fourier description in the circumferential direction are used away from this local zone. In contrast to the earlier work of the authors, the geometric non‐linearity is taken into account in the axisymmetric zone as well. The harmonic coupling in the axisymmetric zone is efficiently handled through the pseudo‐load approach. A special preconditioned conjugate gradient iterative method is employed in conjunction with the arc length method for achieving improved convergence and negotiating the limit points. The attractive features of this methodology are that the tangential stiffness matrix of the structure is never assembled and factorized and that most of the computations are simple matrix–vector multiplications which are carried out efficiently at the element level. Numerical examples are presented to demonstrate the applicability of this method.

偏平シェル理論に基づく低次三角形要素を用いた薄肉弾性体の大変形解析法

In the large displacement analysis of thin shells, a flat triangular finite element has been used for a long time. Such a triangular element is very efficient to analyze a shell with an arbitrary shape numerically. Contraversely, it is impossible to evaluate the effect of the curvature in a shell exactly by the usual flat element. On the other hand, the general shell element which is based on the shell theory, is exact for the reason that it is estimated the curvature, but has been involving important numerical problems, so-called the membrane locking behavior and so on. In this paper, we describe the new large displacement analytical method of shell structures using the triangular finite element proposed by Carpenter et. al. This simple three-node low-order triangular element is baed on Marugerre's shallow shell theory, and is considered having away from the membrane locking behavior. The characteristics of this new numerical analytical method is the formulation technique in applying to the incremental theory. Namely, the local deformed shape of the analysis model is recognized as effect of curvature of the each element like the initial geometrical imperfection, but the global deformed shape of the analytical model is done by updated Lagrangian formulation. At last, we show some numerical examples to examine the validity and efficiency of the present numerical analytical method.

A novel shell element including in-plane torque effect

A novel nine degree-of-freedom flat triangular membrane element is developed by introducing a kinematic degree of freedom associated with rotation about the surface normal at each node. This element is combined with the plate bending element to formulate a general shell element. This in-plane element eliminates the well-known problems caused by zero stiffness corresponding to the in-plane rotation in the shell element. The applicability and the accuracy of the present element are demonstrated through the analysis of various linear and non-linear problems.

A 20-DOF hybrid stress general shell element

A hybrid stress general shell element is developed based on the Hellinger-Reissner principle modified for relaxed element compatibility conditions. The element is based on a thin shell theory with Love-Kirchhoff hypothesis. It is of quadrilateral shape with only four corner nodes and five degrees of freedom per node. The geometry of the element is approximated through a cubic polynomial surface patch. Numerical examples consisting of torsion-loaded slit cylinder and pinched cylinders with open ends and rigid diaphragmed ends demonstrate excellent performance of the present element

Shells of revolution with local plasticity

A nonlinear finite element model that is suitable for the static analysis of shells of revolution with local plasticity is developed. Actually, shells of revolution always exhibit local deviations, like a cutout, a junction, and/or an imperfection. The stress concentration caused around a local deviation may make the material plastic in the surrounding region. The analytical model consists of three different types of shell elements: rotational, general, and transitional. The rotational shell elements are used in the region where the shell is axisymmetrical, and the general shell elements are deployed in the region of the deviation. The transitional shell elements are inserted between the two distinctively different types of elements to achieve continuity of displacement fields. Only the general shell element possesses the material nonlinear properties to capture the localized plasticity. A simple description of plastic behavior based on elastic-plastic behavior, the Von Mises criterion, the Prandtl-Reuss flow rule, and a layered structure, developing plastification through the thickness of the general shell, is used. The stress components at appropriately chosen station points covering the entire volume of the element are stored during the computation. A check is made for initial yielding in the general shell elements at the end of each step of loading. The results of three numerical studies elucidate the localized nonlinear material behavior in a rotational shell structure. In the first example, an axisymmetric junction problem is studied to check the technique against published results. Then, in the last two examples, a cylindrical shell with a circumferential line crack and with a circular cutout are studied. Since the selection of the size of the substructure and the number of harmonics are very important for the localized plasticity problem in the rotational shell, detailed convergence studies are presented. It is shown that the local-global analysis is an attractive alternative to the entirely general element style analysis for axisymmetric shell structures with local imperfections.

A formulation of general shell elements—the use of mixed interpolation of tensorial components

AbstractWe briefly discuss the requirements on general shell elements for linear and nonlinear analysis in practical engineering environments, and present our approach to meet these needs. We summarize and give further insight into our formulation of a 4‐node shell element using a mixed interpolation of tensorial components, and present a new 8‐node element using this approach. Specific attention is given to the general applicability of the elements and their efficient use in practice.

A mindlin element for thick and deep shells

A strain-displacement relation of the Reissner-Mindlin type is derived for a general shell element which may be used with equal confidence and ease in either plate or shell configurations over a wide range of thicknesses. The usual assumptions pertaining to thin, shallow shell geometry have not been employed, resulting in strain-displacement relations which are consistent with the Reissner-Mindlin hypotheses for thick shell configurations. Test results show that this element has excellent accuracy and convergence characteristics and is free of shear locking.

Numerical analysis of electromagnetic stress induced in bellows for a magnetic fusion reactor

The use of bellows in a magnetic fusion reactor gives, in many cases, more efficient penetration of the poloidal electric field and very efficient plasma heating in the first phase of operation. This is because bellows sections lead to higher electrical resistance of the vacuum vessel. There are many examples of its application to tokamak type experimental devices such as TFTR, JT-60 and so on. In addition to its higher electrical resistance, the bellows has an ability to absorb large deformation due to thermal expansion and radiation creep, and thus has been applied to the design of a first wall configuration in some conceptual designs of a commercial power reactor. Its application to a vacuum vessel of the Reacting Plasma machine, which was proposed by the Institute of Plasma Physics of Nagoya University, was examined for circular plasmas. When applying the numerical analysis to the bellows, a computational difficulty occurs. Too many unknows appear in a mesh division with a general shell element and these are not easily handled. Therefore in the present paper a finite element code is made to consider an axisymmetric structure loaded by nonsymmetric forces. In the code various types of electromagnetic forces are taken into account. These include the diamagnetic electromagnetic force, the toroidal-current-induced-electromagnetic forces, the electromagnetic forces due to toroidal coil current change and saddle shaped electromagnetic forces. Aluminum is selected as bellows material because of its low activation nature but there are large induced currents because of its higher electrical conductivity. Results of stress analysis show that a leak of the saddle shaped current into the thick aluminum vacuum vessel generates maximum bending stress near a joint between the bellows and the vessel. Stress due to other types of electromagnetic force are significantly smaller.

Shells of revolution with local deviations

AbstractA finite element model that is suitable for the analysis of shells of revolution with arbitrary local deviations is presented. The model employs three types of shell elements: rotational, general and transitional. The rotational shell elements, which are most efficient, are used in the region where the shell is axisymmetric. The general shell elements, which can simulate almost any shell geometry, are used in the local region of the deviation. The transitional shell elements connect these two distinctively different types of elements and make it possible to combine them in a single analysis.The form of the global stiffness matrix is somewhat unique in the new model. Non‐zero terms are not confined to a narrow band along the diagonal, but occur throughout the matrix. This is due to the following: (1) two different types of nodes, ring nodes and point nodes, are combined in a single analysis; and (2) a locally non‐axisymmetric geometry creates a coupling of the Fourier harmonic coefficients of the rotational elements. Yet, the matrix still contains many scattered zero terms that should be considered for numerical efficiency. In this paper an efficient solution procedure that is effective for this situation is developed. The steps include the use of a substructuring technique and separate partial harmonic analysis. A numerical example is presented and compared with existing solutions to demonstrate the capabilities and the efficiency of the new model.

Shell Elements for Cooling Tower Analysis

Intending to achieve optimum finite element modeling of column supported cooling towers for seismic response studies according to the distributions of dominating bending and membrane stresses and intending to model the vulnerable shell‐column region using discrete column elements and quadrilateral shell elements, a set of elements is adopted, modified or extended. The set includes: (1) A 16 d.o.f. column element; (2) a 48 d.o.f. doubly curved quadrilateral general shell element; (3) a 42 d.o.f. doubly curved general‐membrane transition element; (4) a 21 d.o.f. and 39 d.o.f. doubly curved triangular membrane filler elements; and (5) a 28 d.o.f. doubly curved quadrilateral membrane element. Examples are given to evaluate a single type, combined types, and the whole set of elements with results in good agreement with alternative solutions. These elements may be used for accurate and efficient free vibration analysis of column supported cooling towers. The modeling may be used in seismic response analysis of cooling towers for obtaining detail stress distribution in the vulnerable shell‐column region. This model may be extended to include material nonlinearity in the shell‐column region for the seismic response studies.

Finite element techniques for the analysis of cooling tower shells with geometric imperfections

Two finite element methods for analysing geometrically imperfect cooling tower shells are presented. In the first the geometry of the imperfection is modelled by the elements; in the second the imperfection is represented by an equivalent load on the shell. Axisymmetric and general shell elements have been considered. Results are given which show that the first approximation to the equivalent load is sufficiently accurate and that it is possible to represent local imperfections by axisymmetric imperfections which require less computation. It is also shown that axisymmetric elements should be used wherever possible, because of their greater efficiency, following the geometry of an axisymmetric imperfection but representing local imperfections by equivalent loads.

Line node and transitional shell element for rotational shells

AbstractA transitional shell finite element which is useful for the analysis of shells of revolution with local deviations is developed. For the analysis of such a structure, rotational shell elements may be used in the axisymmetric portion of the shell, while general shell elements are needed in the region where deviations create a local non‐axisymmetry. The transitional shell element connects these two different types of elements. To facilitate the transition, a line node, which can accommodate any type of shape function, is developed. Numerical examples are presented to demonstrate the accuracy of the new element.

Lumped mass matrices with non−zero inertia for general shell and axisymmetric shell elements

AbstractA diagonal lumped mas formulation with non−zero inertia terms is presented for Ahmad's general shell element. The effect of co−ordinate transformation of the mas matrix is demonstrated. Due to arbitray co−ordinate transformation on nodal variables, the diagonal lumped mas matrix becomes a banded matrix of half bandwidth three. It is shown that with some approximations this matrix can be made diagnoal without effecting the results appreciably. Numerical examples are presented to illustrate the accuracy of the formulation. Similar lumped mass matrix formulation is also given for axisymmetric shell elements.