Developing a shell finite element for large displacements of aircraft structural elements modelling


Nizametdinov F. R.1*, Sorokin F. D.1**, Ivannikov V. V.2***

1. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia
2. Scientific and technical centre of rotor dynamic «Alfa-Tranzit», 1, Leningradskaya str., Khimky, Moscow region, 141400, Russia



Structural elements in mechanical engineering, including structural aircraft elements, are often described by a model, for which large generalized displacements at small deformations are characteristic. Thus, the parts, such as aircraft skins sections or aircraft engine housings, while considering evolution; thin-walled rotors etc. can be referred to these elements [1, 2].

Application of a moving coordinate system associated with an aircraft leads to the necessity of introducing extra inertial loads (gyroscopic, Coriolis, et-cetera) while an aircraft evolution considering. In case of deformable systems, such as aircraft hull and wings, it leads serious complications in mathematical description of a finite element. However, if immovable coordinate system, associated with the Earth, is employed, then the necessity for the extra loads does not arise, since the movement is not decomposed into translational and relative components, though large rotations, which an aircraft performs while moving relative to the Earth, should be herewith accounted for.

The purpose of the article consists in developing such a finite element, in which displacements and rotations are large, while deformations and, hence, relative changes of the element size are small.

Many works were devoted, and a whole number of techniques were developed to solve the problem of describing large displacements of structural elements to determine displacements and power factors in them. A rather complete review of these works and techniques is presented in [3, 4]. Particularly, works [3, 4] mentioned the corotational approach with the Euler vector application for finite rotations description, which was employed in this paper. This approach is rather new. Thus, it has not yet reached the same level of popularity as the Lagrange approaches both general and modified [6, 7].

Despite its novelty, the approach is based on the old idea of separating displacement as rigid whole and purely deformational motions. Each approach has its drawbacks. Lagrange approaches have one common disadvantage, which consists in the fact that each element must be recoded from scratch for geometrically nonlinear analysis [8]. The main drawback of the corotation approach is the problem of tracking the shadow position, and selecting the point with which this position is associated. However, there are studies demonstrating that the best choice is binding to the element center of mass [9].

Due to the growing interest in such an approach, many studies devoted to the development of ordinary [10, 11] and specialized [12-15] finite elements using this method appeared in recent years. The main differences of the approaches proposed in this paper are the possibility of obtaining fully analytical relations for the stiffness matrix and the vector of elastic forces; the ability to describe unlimited large rotations, and the absence of the need to additional degrees of freedom introducing.

The Euler vector used in this work is the most common and natural way to describe finite rotations [16-19], though, the Euler vector has a disadvantage, consisting in the presence of a limit value of the rotation angle (2π). With this rotation angle value, the tensors linking small physical rotations with derivatives of kinematic parameters appear degenerate. To overcome the problem, the presented work employs modification based on Euler vector correction while approaching the limiting value. The modification is described in detail in [3] and is not linked to both the base element or to the number of nodes.

The article presents an algorithm for deriving the tangent stiffness matrix and the vector of elastic forces for a geometrically nonlinear shell finite element on the example of a four-node finite element. However, the algorithms are of rather general character and can be applied to almost any basic element (not necessarily a shell) without significant changes.

Based on the test problems solution results, an inference can be drawn on the derived relationships correctness, rather good accuracy of the developed element, which allows its successful application for computing structural elements of machines, subjected to large generalized displacements at small deformations. Besides, the authors can claim that the developed element possess high performance compared to the analogs due to the transparent closed analytical expressions for tangent stiffness matrices and the nodal force vector.


Euler vector, rotation tensor, large displacements, large rotations, tangent stiffness matrix, finite element, shadow element


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