A priori estimation of the systematic error of consistently derived theories for thin structures

Abstract

The consistent approximation approach is a dimension reduction technique for the derivation of hierarchies of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It is based on the uniform truncation of the potential energy after a certain power of a geometric scaling factor that describes the relative thinness of the structural member. The truncation power defines the approximation order N of the resulting theory.In the contribution, we deal with the exemplary case of a slender, beam-like continuum with rectangular cross-section. We introduce an extension of the consistent approximation approach towards a simultaneous, uniform truncation of the complementary energy. We show that compatible displacement boundary conditions can be derived from the Euler-Lagrange equations of the complementary energy, whereas, field equations and stress boundary conditions follow from the Euler-Lagrange equations of the potential energy. Thus fully defined boundary value problems are obtained for all orders of approximation N.We provide an a priori-error estimate for the systematic error of the Nth-order theory. It states that the norm difference of the solution of the so derived boundary value problem of order N to the exact solution of the three-dimensional theory of elasticity declines like the geometric scaling factor to the power of N+1. Thus the approximation character of the approach is proved.Furthermore, we outline why the approach is to prefer over the - in engineering mostly used - approach of a fixed displacement field ansatz with unknown coefficients. We show that the later approach leads in general to theories of higher complexity without increasing the approximation accuracy compared to a consistent approximation.