Strict upper and lower bounds of quantities for beams on elastic foundation by dual analysis

Purpose - The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis. Design/methodology/approach - It leans on the dual analysis wherein the constitutive relation error (CRE) is used to perform goal-oriented error estimation. Due to the coupling of the displacement field and the stress field in the equilibrium equations of the beam, the prolongation condition for the stress field which is the key ingredient of CRE estimation is not directly applicable. To circumvent this difficulty, an approximate problem and the solution thereof are introduced, enabling the CRE estimation to proceed. It is shown that the strict bounding property for CRE estimation is preserved and strict bounds of quantities of the beam are obtainable thereafter. Findings - Numerical examples are presented to validate the strict upper and lower bounds for quantities of beams on elastic foundation by dual analysis. Research limitations/implications - This paper deals with one-dimensional (1D) beams on Winkler foundation. Nevertheless, the present work can be naturally extended to analysis of shells and 2D and 3D reaction-diffusion problems for future research. Originality/value - CRE estimation is extended to analysis of beams on elastic foundation by a decoupling strategy; strict upper bounds of global energy norm error for beams on elastic foundation are obtained; strict bounds of quantities for beams on elastic foundation are also obtained; unified representation and corresponding dual analysis of various quantities of the beam are presented; rigorous derivation of admissible stresses for beams is given.