Keywords: twisted beam, non-linear spatial beam, non-linear initial geometry, reinforced concrete, Möbius band.

The twisted beams are interesting from both, architectural and engineering point-of-view. They are, however, only rarely employed as a load carrying members of reinforced concrete structures. This is in part due to the fact that the tools for the analysis of this kind of structures are not well developed.

The finite element analysis of the twisted cantilever beam was used by several authors for the evaluation of their finite elements. Such an analysis is often presented by researchers to prove that their finite element is capable of considering the initially curved and twisted axis of the beam properly. The following standard test problem for verifying the finite element accuracy, presented by MacNeal and Harder [2], is usually performed: the beam is clamped at one end and loaded either with unit in-plane or unit out-of-plane force at the other; the centroidal axis of the beam is straight at the undeformed configuration, while its cross-sections are twisted about the centroidal axis from 0 at the clamped end to at the free end. The cross-section of the beam is rectangular and the material of the beam is linear. In a slightly greater detail, the problem was analyzed by Tabarrok et al. [1], yet the linear material of the beam is still taken into account.

There are at least two natural kinds of pre-twists. In the first one, the initial twist angle is taken to be the linear function of the arc-length of the line of centroids, and the related twisted shape of the undeformed cantilever has curved edges. Such a shape is typical for various screws and turbine blades. The second kind of a pre-twist assumes the straight edges of the beam which implies a non-linear dependence between the pre-twist angle and the arc-length of the line of centroids. The related geometrical aspects are studied in Zupan and Saje [3]. In this article we will limit ourselves only to the linearly twisted beams.

In the analysis of twisted beams, we employ our strain-based finite-element formulation [4]. This formulation uses the `geometrically exact finite-strain beam theory' of Reissner [5] and Simo [6] and employs the translational and rotational total strain vectors as the only interpolated variables. Such a formulation proved to be suitable for describing finite translations, rotations, and deformations, and a complicated initial geometry of the structure. For the linear material, the efficiency is demonstrated by studying the deformation of a linearly twisted cantilever beam and the split Möbius band subject to an out-of-plane force. These examples indicate a substantial influence of the non-linear initial geometry on the behaviour of the beam. The pre-twist leads to the deformations in directions not parallel to the applied force. Moreover, the twisted beams are generally stiffer than the non-twisted ones.

The increased stiffness of the twisted beams (especially in some directions) is fully due to the different initial geometry. Thus it is of a considerable interest to see how the pre-twist effects the beams made of non-linear materials such as reinforced concrete. We assume the constitutive law proposed by Desayi and Krishnan [7] for concrete in compression, and that of Bergan and Holand [8] for concrete in tension, and focus on columns subjected to an eccentric axial force at one or on both ends. The influence of the pre-twist on the critical force is examined.

1

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

2

R. H. MacNeal, R. L. Harder, "A proposed standard set of problems to test finite element accuracy", Finite Elem. Anal. Design 1, 3-20, 1985. doi:10.1016/0168-874X(85)90003-4

3

D. Zupan, M. Saje, "On A proposed standard set of problems to test finite element accuracy: The twisted beam", Finite Elem. Anal. Design (in press), 2004. doi:10.1016/j.finel.2003.10.001

4

D. Zupan, M. Saje, "Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures", Comput. Methods Appl. Mech. Eng. 192, 5209-5248, 2003. doi:10.1016/j.cma.2003.07.008

5

E. Reissner, "On finite deformation of space-curved beams", J. Appl. Math. Phys. 32, 734-744,1981. doi:10.1007/BF00946983

6

J. C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I", Comput. Methods Appl. Mech. Eng. 49, 55-70, 1985. doi:10.1016/0045-7825(85)90050-7

7

P. Desayi, S. Krishnan, "Equation for the stress-strain curve of concrete", Journal of American Concrete Institute 61, 345- 350, 1964.

8

P. G. Bergan, I. Holand, "Nonlinear finite element analysis of concrete structures", Comput. Methods Appl. Mech. Eng. 17/18, 443-467, 1979. doi:10.1016/0045-7825(79)90027-6

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