Post‐buckling optimization of composite structures using Koiter's method

Abstract

SUMMARYThin-walled structures, when compressed, are prone to buckling. To fully utilize the capabilities of such structures, the post-buckling response should be considered and optimized in the design process. This work presents a novel method for gradient based design optimization of the post-buckling performance of structures. The post-buckling analysis is based on Koiter's asymptotic method. To perform gradient based optimization, the design sensitivities of the Koiter factors are derived and new design optimization formulations based on the Koiter factors are presented. The proposed optimization formulations are demonstrated on a composite square plate and a curved panel where the post-buckling stability is optimized. Copyright c 0000 John Wiley & Sons, Ltd. Thin-walled structures are often designed such that buckling does not occur during service. To ensure a structure does not buckle, the specified buckling load is often much greater than the design load. If a structure can be allowed to buckle during operation, thus operating in the post-buckling regime, it enables the possibility to design lighter and more efficient structures. To enable such an approach, the engineer must optimize the post-buckling response of the structure. Fiber reinforced polymers are ideally suited for such design tasks, as these materials allow a high degree of tailoring of the considered structure, and thus applied here for the post-buckling design optimization of structures. When optimizing structures, robustness of the resulting structure is of major importance, as the imperfection sensitivity can increase during the optimization process [1]. One method is to collect these into an equivalent geometric imperfection and use that to evaluate the knockdown in performance. Refs. [2,3] demonstrate robust design optimization by combining "worst" shape imperfection optimization and laminate optimization, thereby efficiently decreasing the imperfection sensitivity of laminated composite structures. A different approach is to handle all imperfections simultaneously by modeling the uncertainties using statistics, therefore quantifying * Correspondence to: Department of Mechanical and Manufacturing Engineering, Aalborg University (AAU), the imperfections arising from material, geometry, load etc., and perform robust buckling optimization based on the uncertainties, see e.g., [4] for a review of different approaches. Many textbooks describe the coupling between imperfections, buckling load factor, and post-buckling stability, see e.g., [5]. Focusing on simple i.e., distinct buckling load factors, the sensitivity towards imperfections relates to the stability of the post-buckling response. Generally speaking, a stable post-buckled structure is less sensitive towards imperfections than an unstable post-buckled structure. Because of these considerations, the post-buckling stability can be utilized to design robust structures.Post-buckling analysis of plates and shells has been subject to much research. Driven by the aerospace industry, large re...

A novel reduced‐order modeling method for nonlinear buckling analysis and optimization of geometrically imperfect cylinders

“…where the tangent stiffness K t R is calculated by the derivative of the load multiplier vector with respective to the generalized displacement , and its analytical expression can be obtained directly, as given by (13). 2.…”

“…The reduced-order modeling methods [11][12][13][14] proposed based on Koiter perturbation theory 15 offer an alternative to markedly reduce the number of degrees of freedom in large-scale finite element models. The main feature of the Koiter reduction method 16,17 is to construct the reduced-order model at the bifurcation point by representing the displacement field to be an approximate superposition of buckling modes, thus the range of validity of the reduced-order model is limited to the initial postbuckling stage and the prebuckling nonlinearities cannot be considered very well.…”

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

“…However, given the practical significance of Koiter's theory in the problem of structural stability and given the attempts of its FE implementation since early 1970s [9], it is still unavailable from any mainstream commercial FE code, for which disappointment is an understatement. As a result, the research activities on Koiter's theory declined, given sparsely populated publications [10][11][12][13][14][15][16][17][18][19][20][21][22][23] as a representative spectrum of them from various perspectives, without occupying the length of this paper for a comprehensive literature review while a reasonable coverage can be found in relatively recent publications [24,25]. The reality is that for, most structural designers and analysts, Koiter's theory is becoming distant past instead of useful tool at their fingertips.…”

Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling finite element analysis of plates

“…The asymptotic theory originally proposed by Koiter [1] allows a rapid evaluation of the initial post-buckling behavior of structures. The method has been used within a semianalytical context [2,3] and has also been applied within a finite element framework [4,5,6,7,8,9,10,11,12,13,14,15,16]. In recent years, the method has been applied in the context of variable stiffness of panel-type structures [17,18,14,16,19,20] and in the analysis of imperfection sensitive shells [21,22,23].…”