Keywords: beam, three-dimensional beam, three-dimensional rotations, curvature, finite element.

Engineering structures are often modelled using beam models. In the present paper, we limit ourselves to models that are derived from the resultant forms of the differential equilibrium equations. The associated strain-displacement relations consist of three displacement components, and six strain measures (longitudinal and shear strains, and pseudo-curvatures) of the axis of the beam. These strain measures are such that the relationships between the displacements, the strains, and the stress resultants are consistent with the virtual work principle at the deformed state for any magnitude of displacements, rotations and strains. The model is often called the `geometrically exact finite-strain beam theory' (Simo and Vu-Quoc [1]) although both, its exactness (Li [2]), and its applicability to finite strains, when e.g. plasticity material models are used, may be questioned.

Because the spatial rotations are elements of a multiplicative group, the configuration space of deformations is a non-linear manifold. That is what makes the study of these engineering structures so interesting and challenging. The way the rotations are parametrized in the theory is crucial; a particular selection of the parametrization has a direct influence on the algorithm and on the form of the tangent stiffness matrix. In the present work, we employ the rotational vector.

In contrast to previous formulations, which base the finite element implementation of the geometrically exact beam theory on both, displacements and rotations, as the interpolated degrees of freedom (as, e.g., Simo and Vu-Quoc [1], Ibrahimbegovic [3], Crisfield [4]), or solely on rotations (Jelenic and Saje [6]), the present finite element implementation of geometrically exact 3D beam theory parallels the one given in Planinc et al. [5] for plane frames and employs the pseudo-curvature vector as the only degree of freedom that needs to be interpolated along the element. The displacement and rotational components do not need to be interpolated. This `one-field' formulation not only results in the fact that the locking never occurs (that is also characteristic of finite elements by Jelenic and Saje [6]) but results as well in enhanced accuracy for the same number of degrees of freedom compared to displacement-based finite elements, and further enables more realistic description of strain and stress distributions within the beam element.

The stress-resultants as obtained from the equilibrium equations and those calculated from the constitutive equations, do not equal in standard finite element formulations. The corresponding computed error in internal forces may be considerable for materially non-linear problems. This `inconsistency of equilibrium at cross-sections' is here solved by enforcing the consistency condition to be satisfied in a set of predefined points (here taken to coincide with the interpolation nodes) (the `collocation'). A similar strategy was employed by Vratanar and Saje [7] for elastic-plastic analysis of plane frames. However, in the present formulation, the determination of internal forces does not require the differentiation with respect to the arc-length, which is a notable advantage.

The non-linearity of the configuration space of the beam requires a special care in applying Newton's method to the problem. The linearization is made in the sense of the first variation of functionals. That way the variations of the unknowns become equal to the iterative increments of the unknowns. The variations of the unknowns are elements of the tangent space. Thus a special update procedure needs to be applied to map the unknowns from the tangent space onto the configuration one.

A number of finite elements of different order have been tested by various numerical examples. In the paper we consider these problems: (i) a lateral buckling of an in-plane-rigid cantilever, (ii) the in-plane stability of a deep circular arch, and (iii) the bending stiffness of pretwisted cantilevers subjected to the free-end concentrated forces. A rapid (quadratic) convergence has been the characteristics of all elements. An excellent accuracy has been proved by higher-order elements.

1

J.C. Simo, L. Vu-Quoc, "A three-dimensional finite-strain rod model. Part II: Computational aspects", Comput. Methods Appl. Mech. Eng. 58, 79-116, 1986. doi:10.1016/0045-7825(86)90079-4

2

M. Li, "The finite deformation theory for beam, plate and shell. Part III. The three-dimensional beam theory and the FE formulation",Comput. Methods Appl. Mech. Eng. 162, 287-300, 1998. doi:10.1016/S0045-7825(97)00348-4

3

A. Ibrahimbegovic,"On the finite element implementation of geometrically non-linear Reissner's beam theory: 3d curved beam element", Comput. Methods Appl. Mech. Eng. 122, 11-26, 1995. doi:10.1016/0045-7825(95)00724-F

4

M.A. Crisfield, "A consistent co-rotational formulation for non-linear, three-dimensional beam elements", Comp. Meth. Appl. Mech. Engng. 81, 131-150, 1990. doi:10.1016/0045-7825(90)90106-V

5

I. Planinc, M. Saje, B. Cas, "On the local stability condition in the planar beam finite element", Structural Engineering and Mechanics, (in press), 2001.

6

G. Jelenic, M. Saje, "A kinematically exact space finite strain beam model-finite element formulation by generalized virtual work principle",Comput. Methods Appl. Mech. Eng. 120, 131-161, 1995. doi:10.1016/0045-7825(94)00056-S

7

B. Vratanar, M. Saje, "A consistent equilibrium in a cross-section of an elastic-plastic beam",Int. J. Solids Structures 36, 311-337, 1999. doi:10.1016/S0020-7683(97)00327-2

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