Limit analysis of periodic beams

One of the purposes of this paper is to determine Gbhom, the overall homogenized Euler–Bernoulli strength domain of a periodic   rigid-perfectly plastic beam. It also aims at studying the relationship between the 3D and the homogenized Euler–Bernoulli beam limit analysis problems. In the homogenized beam model, the generalized 1D stresses are the axial and the flexural stress field resultants. The homogenization method suggested by [Bourgeois, S., 1997. Modélisation numérique des panneaux structuraux légers. Ph.D. thesis, University Aix-Marseille; Sab, K., 2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. Comptes Redus Mecanique 331, 641–646], and [Dallot, J., Sab, K., 2008a. Limit analysis of multi-layered plates. Part I: The homogenized Love–Kirchhoff model. Journal of the Mechanics and Physics of Solids 56 (2), 561–580] for periodic rigid-perfectly plastic plates is extended to periodic beams. The homogenization procedure is justified using the asymptotic expansion method. Lower and upper bounds for Gbhom are provided. These bounds are analytically obtained in terms of local axial ultimate strengths. Special cases of axially-invariant beams and laminated beams are studied and an improved upper bound is derived. Considering several sandwich beams, it is found that the discrepancy between the lower bound and the improved upper bound is small.