Yan, Jiayi
(2019)
Finite element implementation of Koiter's initial post-buckling analysis.
PhD thesis, University of Nottingham.
Abstract
Thin-wall structures as the main components in the aircrafts, as well as in many other types of structures, such as automotive, trains, ships, etc., are exposed to complex loading conditions. For some parts of these structures, e.g. the upper surface of wings of aircrafts, buckling is the most common mechanism of failure which designers have to ensure their safety with confidence. Prediction of buckling loads as an eigenvalue problem has become a standard provision from most of commercial FE codes as typical design tools nowadays. However, experienced designers and engineers are fully aware of the uncertainties in the structural collapsing load, as some structures could collapse at a load level significantly lower than the predicted buckling load. On the other hand, there are also structures which can sustain increasing loads after buckling to levels much higher than the buckling load. Such phenomenon has been well accommodated and explained by Koiter’s initial post-buckling theory which was established in 1940s. After Koiter’s thesis was translated into English, it quickly gained its popularity in 1960-70s and a lot of applications of it were made mostly by analytical means, whilst its implementation in FEM has been extremely disappointing, although attempts in this direction could be traced as far back as early 1970s shortly after Koiter’s thesis was made known. Most successful applications of Koiter’s theory were made to relatively simple structures due to the limitation of analytical means of analysis. In fact, there has been no lack of publications involving FEM of Koiter’s theory in the past half a century. Nevertheless, the fact is that the theory has not been incorporated in any mainstream commercial FEM code yet.
As an alternative, some commercial FEM codes, e.g. Abaqus, Ansys and Nastran, offer facilities for users to by-pass the bifurcation point in order to obtain the post-buckling deformation after introducing a certain initial imperfection. This is inevitably a highly nonlinear problem and hence computationally expensive. An even more worrying fact is that it might not provide what designers need whilst having potential to mislead designer. In fact, researchers had been there as early as 1940s to look for post-buckling equilibrium path but the endeavour wound up without any major breakthrough soon after Koiter’s theory became known. Through Koiter’s theory, it was understood that the actual buckling behaviour, i.e. whether collapsing prematurely or capable of sustaining higher loads, depends not only on the post-buckling equilibrium path but even more on the stability of the equilibrium state, which is not what a nonlinear analysis could provide. The efforts made to obtain the post-buckling deformation are believed to be on a wrong track. Successful implementation of Koiter’s theory in FEM as a numerical approach would make them redundant, as it did once before when analytical approaches were employed.
In order to achieve this goal, the first task in the present project is to identify the obstacles which have prevented Koiter’s theory from being implemented satisfactorily through FEM. A closer examination suggests that Koiter’s theory has not been incorporated in mainstream commercial FEM codes not without genuine reasons. However, researchers seemed to shy away from revealing the problems directly and explicitly before the difficulties could be addressed appropriately. The objective of this project is to clear these obstacles and to demonstrate fully its feasibility. These key obstacles are found as follows.
1) A key output from Koiter’s theory, initial post-buckling coefficient b has been found mesh sensitive. One would have to endure the mesh sensitivity if conventional finite elements were employed or to employ nonconventional finite elements specially formulated, e.g. having higher order of continuity, to avoid mesh sensitivity. None of them would be attractive to any commercial FE code developer.
2) To evaluate the same initial post-buckling coefficient as mentioned above, a set of special simultaneous linear equations as the governing equation for the secondary perturbation mode have to be solved, where a convincing solution lacks. To facilitate the analysis, an iterative approach was proposed and believed to be employed in all previous FE implementations. It is again unattractive to commercial FE code developers.
3) There is lack of verification for the FE implemented Koiter’s theory. A conventional and effective way of verifying FE approaches has been proposed to compare with available analytical results. There has been no lack of analytical results, e.g. a simply support square plate under biaxial compression, after the developments in 1960-70s. Yet, no comparison has been made to analytical results as far as the applications to plates and shells are concerned. The FE implemented Koiter’s theory has therefore been left in a state where even a basic level of verifications has not been fulfilled. No one has acknowledged the fact in the open literatures.
The first obstacle identified above is directly associated with the very basis of FE formulation. Koiter’s theory was based on the total potential energy (TPE). A natural translation of it led to the N-notation formulation of FEM for the geometrically nonlinear problem. Whilst this was seemingly smooth and straightforward, the N-notation appeared to be a stumbling block, as the stiffness matrices obtained could not be easily expressed explicitly. Even if one managed to do so, they were not unique. As a consequence, the N-notation FE formulation of the geometrically nonlinear problem had been left as a ‘symbolism’, not until recently when this issue has been resolved by revealing the full commutativity of displacements in strain energy density as its intrinsic property and hence in the FE formulation based on it. Based on this, the stiffness matrices can be expressed not only explicitly but also uniquely. In absence of an operational N-notation presentation of Koiter’s theory previously, an alternative was to follow the principle of virtual work (PVW), which was equivalent mathematically, yet significantly different for FE implementation. The alternative expressions of the initial post-buckling coefficients have enjoyed great successes in the development of Koiter’s theory in analytical approaches and they have been taken as the only viable means in almost all attempts of FE implementation. However, it did not come without a price, mesh sensitive being the penalty. The direct symptom was a kind of numerical locking phenomenon which is a purely numerical problem as it has never emerged in analytical solutions. A closer examination would reveal the direct use of stresses in the expression of b. Lower accuracy of stresses than displacements is a generic deficiency to FE which is well-known. The lack of accuracy in stresses has therefore been identified as the reason. Suffice it to say, locking has never emerged in analytical solutions and unconventional finite elements employed to avoid locking followed nothing but usual means of improving the accuracy of stress evaluation. The locking issue, once appropriately identified, has been successfully and completely resolved after re-establishing the N-notation FE formulation. When a and b were expressed and evaluated from the N-notation formulation, numerical locking was no longer present because of its reduced reliance on evaluated stresses. Numerical examples have been shown and the first obstacle is believed to have been overcome satisfactorily.
As for the second obstacle identified, a closed form solution has been found to the secondary perturbation displacement in this project. It is mathematically rigorous and computational efficient. The way of obtaining it has been proven to be probably more rational than the same in analytical approaches, e.g. in terms of the existence of the solution, which was mostly taken for granted previously. A rigorous proof has been provided. It is also applicable to problems involving multiple buckling modes. With the closed form solution, the second obstacle has been satisfactorily resolved.
The third obstacle was perhaps the fatal one for the adoption of Koiter’s theory in commercial FE codes. Without convincing verification, it would be against the basic principles of commercial software. It has been shown that the absence of such comparison was not an oversight of previous researchers and the FE results differed from analytical results widely, if one wished to reveal the problem. After a careful examination, the cause for the difference has been identified. In fact, it was because of the different boundary conditions employed in both analyses. Once the boundary conditions in the FE analysis are prescribed as the same as in the analytical solution, excellent agreement with analytical results can be obtained as have been shown in the thesis.
In order to make comparisons with another highly relevant class of structures, shallow curved panels under axial compression, efforts have been made in the reproduction of an analytical solution. In fact, an analytical solution was available but it was derived under the Love-Kirchhoff (L-M) plate and shell theory. With all available finite elements formulated based on the Reissner-Mindlin (R-M) hypothesis, an update is required in order to make sure that comparisons can be made alike with alike. The analytical results obtained degenerates to L-K solution when transverse shear stiffness is set to infinity, which can be further specialised to that of plates when curvature vanishes, leading the results which have been employed in the verification mentioned above when biaxial loading is reduced to a uniaxial loading condition. By degenerating the obtained analytical results here directly to plates, one finds that the expression of b reproduces that as directly obtained for plates. Any variation of its value will be due to the variation in the buckling load. However, the comparisons obtained between the FE and analytical results can be more relevant when the analytical results are based on the R-M theory.
For curved panels as a slightly more sophisticated case than flat plates, the boundary conditions in various regimes implied in the analytical solution have been unravelled and prescribed to the numerical reproduction. Different from flat plate, in-plane shears emerge in the buckling and post-buckling regimes. Moreover, the tangential displacements obtained from the analytical solution have been prescribed as essential boundary condition in order to precisely determine the buckling modes and the secondary perturbation modes. Excellent agreements of values of b have been achieved based on the accurate determination of secondary perturbation modes. While without appreciation of such boundary condition, viz. the normal stresses as the natural boundary condition and tangential displacements as the essential boundary condition, it is impossible to reproduce the analytical results via finite element method.
As a further application of the FE formulation of Koiter’s theory, the mode interaction problem has been analysed. The buckling of a simply supported cylinder under axial compression has been known as a multiple-mode problem. The analytical expressions of Koiter has been resorted to for the detailed illustrations of buckling modes in terms of three displacements for both axisymmetric and non-axisymmetric buckling modes. The boundary conditions hence have been specified and employed accordingly. The numerical buckling loads show excellent agreement with the analytical ones along with the buckling modes. It can be found that the eigenvalues are clustered within a small range of the buckling load, rather than exactly repeated eigenvalues. By using the eigenvectors obtained for all modes within the cluster, the initial post-buckling analyses have been carried out in an approximate sense of mode interaction. Out of the analysis, the severity of the effects of mode interaction on the initial post-buckling behaviour has been demonstrated.
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