A flexible rod finite element with separate storage of cumulated and extra rotations for large displacements of aircraft structural parts modeling

Deformable body mechanics


Popov V. V.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

*e-mail: vvpopov@bmstu.ru
**e-mail: sorokinfd@bmstu.ru
***e-mail: vvivannikov@alfatran.com


A great number of aircrafts parts, modeled numerically, can be treated as a flexible rod, which exhibits large displacements and rotations, but which strains remain small. Spars wing panels, fuselage stringers, screw blades are among these parts.

The paper presents derivation of tangent stiffness matrix and deformation loads vector of a geometrically nonlinear flexible rod finite element with large rotations and small increments stored separately. The tangent stiffness matrix and the deformation loads vector are both written in the closed forms and can be programmed relatively easily. The element is derived with conventional finite rotations theory, based on the Euler vector and rotation tensor. Following the Updated Lagrangian formulation, the element rotations are decomposed into to the accumulated part and the small incremental one. This approach allows avoid possible singularities and reach rotation angle magnitudes as large as 2p and even more.

The proposed algorithms for the stiffness matrix and deformation loads vector computation are verified by some classical benchmarks. The thorough comparison of the obtained numerical results with existing publications, presenting alternative nonlinear beam formulations, reveals the high accuracy and numerical stability of the developed finite element.

It is worth noting, that in the majority of works in this area the authors derive their nonlinear beam models departing from general expressions of the elasticity theory, which are subjected to certain assumptions regarding the dimensions and kinematics of the analyzed object. Such approach though being general (can be applied not only to beams), suffers from excessive complexity for one’s understanding and can hardly be implemented straightforwardly. In the current contribution an alternative paradigm is proposed: the finite element is constructed on the basis of a well-known linear ancestor, which operates with simple strength of materials hypotheses. The classical beam stiffness matrix is wrapped by specific mathematics responsible for the proper description of large rotations and displacements. In this aspect, the developed finite element reminds the corotational FEA concept, described in [28].

Another important advantage of the presented nonlinear beam formulation is the symmetry of its tangent matrix, which is particularly important for numerical implementation, since it permits the more efficient linear equations solvers to be invoked and significantly improves convergence of Newton-Raphson iterations [29]. Not all the geometrically nonlinear beam models (see, for instance, [30]) possess this property.


flexible rod, finite elements, Euler vector, rotation tensor, large displacements, large rotations


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