In this work, a geometrically exact, fully nonlinear Euler-Bernoulli beam formulation is presented. Herewith a special case of Timoshenko rod models, discussed in various publications such as [2] or [4], is considered. Following the basic assumptions of transversal shear rigidity and plane cross sections, the formulation is based on displacements, their derivatives and a torsional rotation angle, in order to cover finite deformations and rotations. A straight reference configuration is considered, whereas the possibility for initially curved configurations is discussed in [1]. Within the Euler-Bernoulli theory a cross section undergoes ridged body translation and rotation. While the translation is described by the motion of the beam axis only, the rotation is accounted within a rotational field, see e.g. [3]. In the present formulation, the rotational field is parametrized by the rotation tensor based on Rodrigues formula, which yields a simple update scheme as shown in [5], and is the first of its kind for Euler-Bernoulli beam models. Using this parametrization an ansatz is presented, that allows for consistent connection of structural members, even in cases of physical discontinuities. On the element level C 1 continuous Hermite polynomials are used to interpolate the displacements an their derivatives, while a Lagrangian interpolation scheme is chosen for the torsional rotation. A numerical Benchmark problems is presented to demonstrate the performance of the finite element formulation.
This work presents geometrically exact shear-rigid rod and shell formulations. Displacements and rotations are finite. Linear elastic constitutive equations for small strains are considered in the numerical examples for the rods. A Neo-Hookean material is considered for the shell. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod, and a flat reference configuration the shell. Consequently, the use of convective non-Cartesian coordinate systems is not necessary, and only components on orthogonal frames are employed. The parameterization of the rotation field is done by the rotation tensor with the Rodrigues formula, which makes the updating of the rotational variables very simple. The usual Finite Element Method was used and C1 continuity is achieved within the element. This method is used to discretize the potentials on a computational domain in terms of the nodal degrees of freedom. Bearing in mind that the potential is nonlinear a Newton-Raphson iteration scheme is chosen to solve this problem. A set of numerical benchmark examples illustrates the usefulness of the formulation and its numerical implementation. These problems were performed and presented satisfying results. Hence, it can be concluded that this formulation shows great promises to be extensively used for general 3D problems for slender structures. Shear-rigid theories can be widely applied on engineering problems. They can be used in oil drilling rods, robot arms and for rib-reinforced shells that are common in aerospace and automobile industry.
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