Dimensional reduction for the beam in elasticity on a bounded domain

The dimensional reduction method is investigated for solving two-dimensionally linear, isotropic elasticity (Δu + (1 + λ/μ) • u + u = 0) on domain Ωd := Ω × (−d,d) R2 by semidiscretization techniques in the transverse direction, where σ > 0 and Ω = (−a,a). The modelling error between the exact solution u and the dimensionally reduced solution un in E1 (Ωd) is precisely obtained as d and n are fixed, where the norm ʹ• ʹE1 is equivalent to the usual norm ʹ• ʹH1. Numerical examples are presented.