Keywords: critical buckling loads, cracked Timoshenko columns, perturbation method.

The analysis of the stability of columns is a topic of great interest in civil, mechanical, aeronautical, nuclear, and offshore fields. It is well known that the presence of cracks diminishes the stiffness of the structure, leading to higher displacements for the same loads and a decrease in buckling loads, one of the most usual modes of instability of column-like structures.

A widely used method to analyse the mechanical behaviour of damaged Euler-Bernoulli beams is to consider them as two beams connected at the cracked section by a rotational spring whose stiffness is related to the crack size and the geometry of the cross section [1]. For the case of Timoshenko beams, where the effects of shear deformation and rotary inertia are non negligible, a discontinuity in the transverse deflection at the cracked section must be also considered.

For the case of Euler-Bernoulli weakened columns, the exact buckling-load values of weakened columns with several end conditions were determined, using the concept of rotationally restrained junction [2]. Loya et al. [3] proposed first-order closed-form expressions derived from the perturbation method, for critical buckling loads of the Euler-Bernoulli column, procedure used previously to calculate the natural frequencies of bending vibrations of Timoshenko beams [4]. For the case of Timoshenko cracked columns, the exact buckling-loads values can be obtained using the procedure presented by Arboleda-Monsalve et al. [5].

In this work, the perturbation method is used to derive a first-order closed-form expression to calculate the critical buckling loads of a weakened Timoshenko column with different boundary conditions, concluding that the buckling loads depends directly on bending moments and shear forces transmitted by the cracked section.

1

L.B. Freund, G. Herrmann, "Dynamic fracture of a beam or plate in bending", Journal of Applied Mechanics, 98, 112-116, 1976. doi:10.1115/1.3423760

2

C.Y. Wang, C.M. Wang, T.M. Aung, "Buckling of a weakened column", Journal of Engineering Mechanics, 130, 1373-1376, 2004. doi:10.1061/(ASCE)0733-9399(2004)130:11(1373)

3

J.A. Loya, G. Vadillo, J. Fernández-Sáez, "First-order solutions for the buckling loads of Euler-Bernoulli weakened colums", Journal of Engineering Mechanics, 136(5), 674-679, 2010. doi:10.1061/(ASCE)EM.1943-7889.0000103

4

J.A. Loya, L. Rubio, J. Fernández-Sáez, "Natural frequencies for bending vibrations of Timoshenko cracked beams", Journal of Sound Vibrations, 290, 640-663, 2006. doi:10.1016/j.jsv.2005.04.005

5

L.G. Arboleda-Monsalve, D.G. Zapata-Medina, J. Dario Aristizabal-Ochoa, "Stability and natural frequencies of a weakened Timoshenko beam-column with generalized end conditions under constant axial load", Journal of Sound and Vibration, 307, 89-112, 2007. doi:10.1016/j.jsv.2007.06.059

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