On the mathematical justification of the consistent-approximation approach and the derivation of a shear-correction-factor free refined beam theory

Abstract

The "consistent approximation" technique is a method for the derivation of analytical theories for thin structures from the settled three-dimensional theory of elasticity. The method was successfully applied for the derivation of refined plate theories for isotropic and anisotropic plates. The approach relies on computing the Euler-Lagrange equations of a truncated series expansion of the potential energy. In this thesis we extend the approach given in Kienzler (2002) towards the simultaneous truncation of a series expansion of the dual energy. The computation of the Euler-Lagrange equations of the truncated series expansion of the dual energy ensures a rigorous derivation of compatible boundary conditions. The series expansions of both energies are gained by Taylor-series expansions of the displacement field. We show that the decaying behavior of the energy summands is initially dominated by characteristic parameters that describe the relative thinness of the structure. Consequently, the energy series are truncated with respect to the power of the characteristic parameters. For the case of a homogeneous, one-dimensional structural member with rectangular cross-section we proof an a-priori error estimate that provides the mathematical justification for this method. The estimate implies the convergence of the solution of the truncated one-dimensional problem towards the exact solution of three-dimensional elasticity as the thickness goes to zero. Furthermore, the error of the Nth-order one-dimensional theory solution decreases like the (N 1)th-power of the characteristic parameter, so that a considerable gain of accuracy could be expected for higher-order theories, if the structure under consideration is sufficiently thin. The untruncated one-dimensional problem is equivalent to the three-dimensional problem of linear elasticity. We prove that the problem decouples into four independent subproblems for isotropic material: a rod-, a shaft- and two orthogonal beam-problems. A unique decomposition of any three-dimensional load case with respect to the direction and the symmetries of the load is introduced. It allows us to identify each part of the decomposition as a driving force for one of the four (exact) one-dimensional subproblems. Furthermore, we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials. Since all propositions are proved for the exact one-dimensional problem, they also hold for any approximative Nth-order theory. The approach is applied to derive a new second-order beam theory for isotropic material free of a-priori assumptions, which in particular does not require a shear-correction. The theory is in general incompatible with the Timoshenko beam theory, since it contains three in general independent load resultants, whereas Timoshenko's theory only contains one. Furthermore, Timoshenko's theory ignores any effects in width direction. However, the assumption of a simple load case allows for a vis-a-vis comparison of both differential equations and in turn, two shear-correction factors for the use in Timoshenko's theory can be derived.