Keywords: nonlinear beam theory, finite element method, three-dimensional rotation, rotational invariant, Newton's method on nonlinear space, invariant preserving algorithm.

In the paper we limit ourselves to the three-dimensional beam models that are derived from the resultant forms of the differential equilibrium equations. The model, often called the `geometrically exact finite-strain beam theory' (Simo and Vu-Quoc [1]), introduces six strain measures: longitudinal and shear strains, and pseudo-curvatures.

Geometry of the three-dimensional beam is described by the line of centroids of cross-sections and by the family of the cross-sections not necessarily normal to the line of centroids at deformed state; therefore, the configuration space of the beam consists of (i) the linear space of position vectors of the line of centroids and (ii) the nonlinear space of rotations of cross-sections. Spatial rotations are elements of a multiplicative group, which makes the three-dimensional theories so demanding.

The crucial part of any formulation is the choice of the primary variable(s) for the finite element formulation. Earlier formulations base the finite element implementation on both, displacements and rotations, as the interpolated degrees of freedom (as, e.g., Simo and Vu-Quoc [1], Ibrahimbegovic [2], Crisfield [3]), or solely on rotations (Jelenic and Saje [4]). In these approaches, the rotations or/and their increments are directly interpolated, neglecting the fact that the rotations are physically non-additive quantities. As reported by Crisfield and Jelenic [5], such an approach leads to non-objective strain measures, when they are calculated from rotations and displacements. Crisfield and Jelenic [5] suggest a strain-objective formulation, yet it requires the construction of special interpolating functions, which can be a complicated task for higher-order finite elements.

In contrast to such formulations, we here assume that strain measures, i.e. membrane strains and the pseudo-curvatures, are primary interpolated variables. In order to apply the strain measures as basic variables, we follow the work by Planinc et al. [6] and extend it to three-dimensional frames by proposing a modified principle of virtual work in which the strain vectors need only to be interpolated along the element.

The fundamental problem of such a formulation is the integration of rotations from the given interpolated pseudo-curvatures. In the planar case, the derivative of rotations with respect to the natural parameter of the line of centroids equals the pseudo-curvature. In three dimensions, an additional transformation matrix, dependent on rotations, needs to be taken into account. The solution of the system of differential equations, which links the rotations and the pseudo-curvatures, can not be found analytically. Tabarrok et al. [7] proposed the integration of incremental changes of pseudo-curvatures in order to obtain the increments of rotations. Such an integration can be performed analytically due to the non-demanding relationship between the incremental vectors. However, in such a formulation, the total rotations and the total curvatures may not satisfy the exact kinematic equations. Schulz and Filippou [8] propose a similar solution, while the interpolation functions taken are analogous to those in Crisfield and Jelenic [5].

In sharp contrast to the above mentioned authors, our solution employs the exact relationship between the pseudo-curvatures and the rotations. Due to the complicated form of the system of differential equations, the numerical method has been used for the integration of the total rotations from the (iterative) pseudo-curvatures. Both strain vectors are interpolated. Because of the additivity of strains, no special, non-additive procedure is needed. The linearity of the basic variables - the strains - simplifies Newton's iteration method and leads automatically to the objectivity of the interpolated strains. The formulation proved excellent in several demanding numerical examples.

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

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

3

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

4

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

5

M. A. Crisfield, G. Jelenic, "Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation", Proc. Roy. Soc. London A 455, 1125-1147, 1999. doi:10.1098/rspa.1999.0352

6

I. Planinc, M. Saje, B. Cas, "On the local stability condition in the planar beam finite element", Structural Engineering and Mechanics 12(5), 507-526, 2001.

7

B. Tabarrok, M. Farshad, H. Yi, "Finite element formulation of spatially curved and twisted rods", Comput. Methods Appl. Mech. Eng. 70, 275-299, 1988. doi:10.1016/0045-7825(88)90021-7

8

M. Schulz, F. C. Filippou, "Non-linear spatial Timoshenko beam element with curvature interpolation", Int. J. Numer. Methods Eng. 50, 761-785, 2001. doi:10.1002/1097-0207(20010210)50:4<761::AID-NME50>3.0.CO;2-2

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