Asymptotic modelling theorems for the static analysis of linear elastic structures

The work presented here justifies the use of restraints with large positive and negative stiffness values to asymptotically model geometric constraints of a structure in linear structural analysis. Replacing constraints by very stiff restraints improves the versatility of variational methods such as the Rayleigh-Ritz method as the limitation on the choice of functions is removed. Based on recently published theorems on the existence of natural frequencies of systems with artificial restraints of positive and negative stiffness and their convergence towards the frequencies of the corresponding constrained systems, a proof is given to show that the displacement of a constrained structure caused by any action along its direction is approached and bracketed by the displacement of systems with artificial restraints of positive and negative stiffness (positive and negative penalty functions) as the magnitude of stiffness is increased. A procedure for the practical use of positive and negative penalty functions in static analysis of linear structures is also proposed.