Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure

Abstract

To cite this version:Saber El Arem. Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure. European Journal of Mechanics -A/Solids, Elsevier, 2009Elsevier, , 28 (6), pp.1079Elsevier, -1087 Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure Saber El AremLaboratoire Sols Solides Structures-Risques, Domaine Universitaire BP53 38041Grenoble, France AbstractThe main purpose of this paper is to complete the works presented by Andrieux and Varé [2002] and El Arem et al. [2003] by taking into account the effects of shearing in the constitutive equations of a beam cracked section in bi-axial flexure. The paper describes the derivation of a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. Properties of the potential energy and convex analysis are used to reduce the three-dimensional computations needed for the model identification, and to derive the final form of the elastic energy that determines the nonlinear constitutive equations of the cracked transverse section. We aim to establish a relation of behavior between the applied forces and the resulting displacements field vectors, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis. The approach has been applied to the case of a cracked beam with a single crack covering the half of its circular cross section.

Nonlinear analysis, instability and routes to chaos of a cracked rotating shaft

“…The key idea of the EDF−LMS approach, as described in many articles (Andrieux and Varé [2], El Arem [9]), is the identification of the characteristics of a beamlike model (Fig.2(c)) from accurate three-dimensional (3D) finite element model, cf. Fig.2(a).…”

“…2 The cracked shaft model of [2] lumped (zero length) nonlinear bending spring ( Fig.2(d)) whose constitutive equations give a complete description of the crack breathing mechanism. It was demonstrated by El Arem [9] that the shearing effects on the breathing mechanism of the crack are negligible and, thus, will not be considered in what follows. The cracked 3D structural element of Fig.2(a) is subjected to an end moment M 2L = (M x (2L), M y (2L)) at z = 2L.…”

“…It is deduced from the compliance function s that was identified using 3D numerical simulations. The relation between k and s is established by the Légendre-Fenchel transform relating w to w * as detailed in [2,8,9]. The constitutive equations are obtained by differentiating w with respect to [θ ] as:…”

“…It opens totally (lowest stiffness) at ϕ = 3π 2 . This approach was first presented by Andrieux and Varé [2] and experimentally validated by Stoisser and Audebert [30] then largely discussed and improved in [8,12,9]. This variation of the stiffness of the nodal element representing the cracked transverse section will be, in the next part of this paper, adapted to be used in a two-Degrees-Of-Freedom (2−DOF) dynamical system describing a rotating shaft with a cracked transverse section.…”

Rigid finite element model of a cracked rotor

“…However, the stiffness and damping for other modes (bending and torsional) of the shaft deflections were not included. El Arem [24] suggested a model in which the transverse cracked section was replaced by two lumped nonlinear flexural and shearing springs. He considered a 3D finite element model of a shaft in which the opening and closure of the crack were governed by the normal stress on the crack lips.…”