Stability studies for curved beams

Abstract

The paper presents a concise framework investigating the stability of curved beams. The governing equations used are both geometrically exact and fully intrinsic; that is, they have no displacement and rotation variables, with a maximum degree of nonlinearity equal to two. The equations of motion are linearized about either the reference state or an equilibrium state. A central difference spatial discretization scheme is applied, and the resulting linearized ordinary differential equations are cast as an eigenvalue problem. The scheme is validated by comparing predicted numerical results for prebuckling deformation and buckling loads for high arches under uniform pressure with published analytical solutions. This is a conservative system of forces despite their being modeled as distributed follower forces. The results show that the stretch-bending coupling term must be included in order to accurately calculate the prebuckling curvature and bending moment of high arches. In addition, the lateral-torsional buckling instability of curved beams under tip moments is investigated. Finally, when a curved beam is loaded with nonconservative forces, resulting dynamic instabilities may be found through the current framework.

Unified Approach for Accurate and Efficient Modeling of Composite Rotor Blade Dynamics The Alexander A. Nikolsky Honorary Lecture

“…Such equations also have been shown to be well suited for the analysis of a wide variety of beam dynamics problems (Refs. [121][122][123].…”

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

“…Hodges (2003) has introduced a set of complete intrinsic equations for the dynamics of initially curved and twisted geometrically exact beams and has shown the advantages of fully intrinsic formulation for a beam under non-conservative transverse follower force. The latter approach which is based on a finite difference scheme has been later used by Chang and Hodges (2009a) and Chang and Hodges (2009b) for the free vibration and stability analysis of curved beams. Khouli et al (2009) and Ghorashi and Nitzsche (2009) also have used finite difference schemes for the spatial discretization of the intrinsic formulation presented by Hodges (2003) for the dynamic analysis of helicopter rotor blades.…”

Effects of approximations on non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches

“…A number of studies of in-the plane buckling and postbuckling behavior of arches have been reported and most of them focus on the circular arches [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The analytical solutions for the in-plane buckling of high circular arches were obtained by Hodges [1] and verified to be accurate later by Chang and Hodges [2] using a finite element (FE) analysis, while the analytical solutions for the non-linear in-plane buckling and postbuckling of shallow circular arches under a uniform radial load and under a central concentrated load were derived by Pi et al [3] and Bradford et al [4] respectively and the solutions were also verified by the FE analyses.…”

“…In deference to the case of circular arches [1][2][3][4], it is difficult to formulate the accurate non-linear longitudinal normal strain for parabolic arches and consequently it is also difficult to derive the accurate analytical solutions for the non-linear in-plane buckling of parabolic arches. Hence, approximate analytical solutions were sought by several researchers [17,18,[31][32][33][34] using an approximation assumption that dy dz 2 ( 1 ð3Þ…”