Asymptotic dimension reduction of a Robin-type elasticity boundary value problem in thin beams

Abstract

We consider a linear elasticity boundary value problem in a beam with Robin boundary condition at an end and on a segment of the lateral boundary in the middle of the beam. The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard asymptotic approach with an additional expansion suggested to fulfil the Robin conditions. The 3D Robin conditions result into 1D Robin boundary conditions for corresponding ODEs. The asymptotic error is estimated and illustrated by a numerical comparison of the 3D and 1D solutions.