Keywords: two-node element, exact formulation, finite element method, flexibility method, linear analysis, shear locking, tapered beam, Timoshenko's beam theory.

Timoshenko's beam theory is an extension of the Euler-Bernoulli beam theory to allow for the effects of transverse shear deformations. Stiffness based finite element methods have some assumptions which depend on the shape of the variation of unknowns. It should be noticed that the actual physical behaviour of elements may not conform to these assumptions. For the static cases, the flexibility formulation is an excellent approach for the analysis of planar frames, since it renders exact element flexibility and stiffness matrices [1].

Backlund [2] has presented a hybrid finite beam element for the analysis of planar frames based on flexibility methods which could analyze the beams with variable cross sections. Thereafter, an exact force formulation for the analysis of planar frames was described by Carol and Murcia [3]. Then, the stiffness matrix for curved beams with variable cross sectional members was presented by Molins et al. [4] using an extension of a flexibility approach. Moreover, Attarnejad and Valipour [5] have suggested a general method for deriving the stiffness matrix of non-prismatic curved Euler-Bernoulli beam elements based on the principle of virtual work.

An innovative Timoshenko beam element is presented in this paper. This formulation is attained by the exact solution of equilibrium equations based on the cantilever beam as a basic statically determinate configuration which leads to the computation of an exact stiffness matrix of the beam elements which is free of locking effects. Transverse applied loads and initial stresses and strains are included in the formulation which allows the analysis of beam elements in every predictable condition. The simplicity of the application of the proposed method to tapered beam elements is described and primary parameters are computed for such beam elements. Exact solutions arising from a lack of additional assumptions in the displacement domain and using longer straight two-node elements are the most important advantages of the proposed method. Finally, some numerical examples are analyzed using the formulation developed and these results compared with other studies. Convenient application, accuracy and requirement of less time are the other advantages of this element.

1

A. Neuenhofer, F.C. Filippou, "Geometrically nonlinear flexibility-based frame finite element", J. Structural Engineering (ASCE), 124(6), 704-711, 1998. doi:10.1061/(ASCE)0733-9445(1998)124:6(704)

2

J. Backlund, "Large deflection analysis of elasto-plastic beams and frames", Int. J. Mechanical Science, 18, 269-277, 1976. doi:10.1016/0020-7403(76)90028-X

3

I. Carol, J. Murcia, "Nonlinear time-dependent analysis of planar frames using an exact formulation - I. Theory", Computers and Structures, 33(1), 79-87, 1989. doi:10.1016/0045-7949(89)90131-4

4

C. Molins, P. Roca, A.H. Barbat, "Flexibility-based linear dynamic analysis of complex structures with curved-3D members", Earthquake Engineering and Structural Dynamics, 27, 731-747, 1998. doi:10.1002/(SICI)1096-9845(199807)27:7<731::AID-EQE754>3.3.CO;2-T

5

R. Attarnejad, H.R. Valipour, "Exact stiffness matrix of non-prismatic Euler-Bernoulli curved beams", Journal of Faculty of Engineering, University of Tehran, 35(1), 35-46, 2001.

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