Asymptotic decomposition in the problem of joined elastic beams



The analysis of problem of joined elastic beams is presented in comparison with the engineering and asymptotic approaches. Our analysis is based on three-dimensional elasticity theory model and recently developed method of local perturbation (Gaudiello and Kolpakov, 2011), which seems to be an effective tool for analysis of fields in the vicinity of joint. We demonstrate that the method of local perturbation developed in (Gaudiello and Kolpakov, 2011) for scalar Laplace equation can be modified for vectorial elasticity theory problem. We demonstrate that the elasticity theory problem in joined domains of small diameter can be decomposed into one-dimensional problem describing global deformation of a system of joined beams and three-dimensional problems describing local deformation of singular joints in uniform fields. The first problem is the classical one, which ignores individual properties of joint absolutely. The second problem initiates reminiscence about the cellular problem of the homogenization theory for periodic structure. In spite of some similarities, the mentioned problems differ significantly. In particular, the joint of normal type (the joint similar in dimensions and material characteristics to the joined beams) does not manifest itself on global level. Due to the strong localization of perturbation of solution, computation of local strains and stresses in the vicinity of joint can be realized with standard FEM software.