Stability of beams with discrete lateral restraints

The current work analyses the lateral stability of imperfect discretely-braced steel beams using variational methods. To facilitate the analysis, Rayleigh-Ritz approximations are used to model the lateral deflection and the angle of twist. The applicability of the methods is initially demonstrated for the cases of unrestrained and continuously restrained beams by comparison with both analytical and numerical solutions of the governing differential equations of the respective systems. The method is then applied in full to the case of a discretely-braced beam. Initially, it is assumed that the degrees of freedom (DOFs) can be represented by single harmonics; this is then compared to the more accurate representation of the DOFs as full Fourier series. After carrying out a linear eigenvalue analysis of the system, it is found that the beam can buckle into two separate classes of modes: a finite number of modes, equal to the number of restraints provided, which involve displacement of the restraint nodes and interaction between distinct sets of harmonics, and an infinite number of single harmonic internodal buckling modes where the nodes remain undeflected. Expressions are derived for the elastic critical moment of the beam, the forces induced in the restraints and the threshold stiffness, i.e. the minimum stiffness required to enforce the first internodal buckling mode, whereupon the beam attains its maximum achievable critical moment. The analytical results for the critical moment of the beam are validated by the finite element program LTBeam, while the results for the deflected shape of the beam are validated by the numerical continuation software Auto-07p, with very close agreement between the analytical and numerical results. Design formulae, from which practical design rules can be developed, are given for the critical moment, restraint force and threshold stiffness. The design rules return values close to those predicted from theory. When compared against equivalent design rules developed based on analogies with column behaviour, it is found that the column rules are generally overly conservative for restraints attached close to the compression ange and considerably unsafe for restraints attached close to the shear centre