Quadrilateral Membrane Element Research Articles

Membrane quadrilateral finite elements with rotational degrees of freedom

A couple of new membrane elements with rotational degrees of freedom are presented. MQ2 is a quadrilateral membrane element with quadratic displacement interpolation and MQ3 is a quadrilateral with cubic displacement interpolation. Mid-side tangential displacements are used as the incompatible mode interpolation parameters to improve the elements' performance in the distorted configuration. Both membrane elements are based on a sound variational foundation and both exhibit an excellent performance over a set of problems.

A 4‐node quadrilateral membrane element with in‐plane vertex rotations and modified reduced quadrature

An assessment is made of a 4‐node quadrilateral membrane element with one rotational and two translational degrees of freedom per node, as formulated by Taylor and Simo. The element, QC9, is formed by degeneration of the 9‐node Lagrange element and condensation of the centre degrees of freedom. An 8‐point, modified reduced integration scheme is implemented in this element, QC9(8), to improve on the 3 × 3 quadrature performance, yet avoid the additional rank deficiency due to reduced integration (2 × 2). QC9(8) performs as good or better than all elements surveyed. It is shown that a similar degeneration of the 16‐node Lagrangian element can be carried out, but that the resulting element fails the patch test.

Zen and the art of finite element design

The paper reviews Robert M. Pirsig's ideas concerning the concept of quality as expressed in his book Zen and the Art of Motorcycle Maintenance and applies them to the problem of designing finite elements. Using quadrilateral membrane elements as a typical example, it is conclude that finite elements differ remarkably in the quality of their performance and that improvements in quality occur in response to the dedication and insight which finite element designers brings to their task.

On the Convergence Properties of the Quadrilateral Elements of Sander and Beckers

A class of nonconforming quadrilateral membrane elements, introduced by Sander and Beckers, is discussed in all details. It is proved that two elements that do not pass the patch test may still yield convergence under a suitable condition on mesh subdivisions, whereas one such element is found to be divergent for all mesh subdivisions. All other finite elements in this class that pass the patch test are convergent. The mathematical analysis provides a clear explanation of the convergence behavior appearing in the numerical examples of Sander and Beckers.

On the convergence properties of the quadrilateral elements of Sander and Beckers

A class of nonconforming quadrilateral membrane elements, introduced by Sander and Beckers, is discussed in all details. It is proved that two elements that do not pass the patch test may still yield convergence under a suitable condition on mesh subdivisions, whereas one such element is found to be divergent for all mesh subdivisions. All other finite elements in this class that pass the patch test are convergent. The mathematical analysis provides a clear explanation of the convergence behavior appearing in the numerical examples of Sander and Beckers.

On the suppression of zero energy deformation modes

AbstractBased on the Hellinger‐Reissner principle and the deformation energy due to assumed stresses and displacements, the problem of the kinematic deformation modes in assumed stress hybrid/mixed finite elements has been examined. Basic schemes are developed for the choice of assumed stress terms that will suppress all kinematic deformation modes. Quadrilateral membrane and axisymmetric elements, and three‐dimensional hexahedral elements, are used to illustrate the suggested procedure.

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.

New thoughts on stropping cracks which emanate from holes

The present paper is concerned with the repair of cracks emanating from holes using as an illustrative example a cracked lug repaired with a bonded steel sleeve. The geometry of the lug is shown in Fig. 1. Two different crack lengths were considered, viz. 1.59 ram and 3.18 mm, and the lug was taken to be a 7079-T6 aluminum alloy. The lug was loaded via a rigid pin and the resultant load distribution around the hole was obtained using the method described in [4,5]. The stress intensity factor at the crack tip was then obtained using the finite element method. To this model of the cracked lug was then added a finite element model of a bonded steel insert. The steel was assumed to have Young's modulus of 206 x l0 v MPa and a Poiss~n's ratio of 0.32 while the adhesive had a Young's modulus of 2.5 x l0 MPa and a Poisson's ratio of 0.32. The resultant finite element mesh consisted of 814 nodal points, 316 linear strain triangles, 7 constant strain triangles, 32 of the four noded quadrilateral membrane elements and one special crack tip element. The effect that this bonded sleeve has on the stress intensity factor K. is shown in Table 1 Here 1 KI /K. u is the ratio of the stress intensity factors after and before t~ sJeeve is inserted while ~A/P and oA/P are the non-dimensionalized values of the maximum shear stress and the maximum compressive stress developed in the adhesive bond respectively. Here A is the internal surface area of the hole and P is the total load applied to the lug. From Table 1 we see that, with a typical adhesive thickness of 0.1 mm, reductions in the stress intensity factor of approximately 50 percent are obtained for both crack lengths. This is a substantial reduction and would lead to a marked slowing down of cracks under a fatigue loading. Let us now turn our attention to the 1.59 mm crack with an adhesive thickness of 0.3 mm and a steel sleeve 3 mm thick. If we perform the coordinate transformation x' = x/3 and y' = y/3 then in this coordinate system the hole is II.3 mm in diameter, the crack is 0.53 mm in length while the adhesive is 0.1 mm thick and the sleeve is 1 mm thick. This geometry is typical of fastener holes in aircraft structures, such as wing spars. The present analysis shows us that a bonded insert for this size hole in a lug reduces the stress intensity factor by 64 percent for a 0.53 mm crack. This crack size is important since it represents the minimum size crack that can accurately be found using present eddy current detection methods. Consequently,

Four‐node quadrilateral stress membrane element with rotational stiffness

AbstractA four‐node quadrilateral membrane element with 12 degrees‐of‐freedom and based on stress assumptions is presented. The element has a rotational degree‐of‐freedom at each node associated with a moment.

An accurate and economical eight‐node stress membrane element

AbstractAn accurate and economical eight‐node quadrilateral membrane element based on stress assumptions is presented. It is shown that this element offers distinct advantages over other available membranes.

Thermal deformation vector for a bilinear temperature distribution in an anisotropic quadrilateral membrane element

AbstractThis paper presents a thermal‐structural deformation vector for an anisotropic quadrilateral membrane element. The elastic properties of the element are based on linear stress assumptions and were developed by Robinson1. A general bilinear temperature field within the element is chosen in order to obtain a thermal deformation capability for the element commensurate with its elastic capability. A pseudo shear term is developed to account for part of the thermal deformation. The element is suitable for finite element analysis of advanced aircraft structures under elevated temperature environments and is particularly well suited for analysis of regions with severe temperature gradients such as occur in the vicinity of local heat sources and sinks. An example problem is included to demonstrate the behaviour of the element.

Effect of out-of-planeness of membrane quadrilateral finite elements.

Investigation of the effect of using plane quadrilateral membrane elements for modeling nonplanar structures. The effect is assessed by analyzing a simplified finite element model with the aid of the structure network analysis program. The results obtained indicate that the use of planar quadrilateral membrane elements for modeling bending problems can lead to large errors if the four points that define the quadrilateral are not in the same plane.