Commutativity of the strain energy density expression for the benefit of the FEM implementation of Koiter's initial postbuckling theory

Abstract

Summary The concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature.

Closed-Form Solution for Secondary Perturbation Displacement in Finite-Element-Implemented Koiter’s Theory

“…where 0 K is the stiffness matrix for the linear problem, and 1 N and 2 N are the stiffness matrices due to geometrical nonlinearity, which are linear and quadratic functions of displacement, respectively, and q 1 is the buckling mode. They can be obtained explicitly and uniquely [26]. 0 c q is defined as follows.…”

“…where 0  K is the so-called initial stress or geometric stiffness matrix with detailed expression provided in [26] after the initial stresses have been normalised by the load parameter .…”

“…Noticing the typographic errors in the above expressions as presented in [26] in the second term in the denominators of ( 7) and ( 8), which have been corrected here. The significance of a and b has been described in the previous section through Fig.…”

“…1) A key output from Koiter's theory, coefficient b, has been found mesh sensitive. While several techniques had been proposed [10][11][12][13], an effective approach has been demonstrated very recently by the authors in [26], simply by adopting the N-notation in FEM.…”

“…In the present paper, a systematic approach along this line will be established leading to a closed-form solution of mathematical rigor. The governing equations will be expressed in this paper using the N-notation finite element formulation as established in [26]. As it is a closed-form solution, no iteration will be required.…”