The numerical studies presented above prove the applicability of the proposed buckling mode identification method. It is also suggested, however, that a relatively fine FEM discretization is necessary if various buckling modes are to be identified, including ones with small buckling lengths. Since the number of cFSM base functions is tied to the FEM mesh, a fine mesh requires a significant number of cFSM base functions, which ultimately results in fairly significant computational effort. This is clearly unfavourable. At the same time, by looking at the details of the participation calculations, one can immediately observe that many of the applied cFSM base functions have negligible contribution in any buckling mode of practical interest, which suggests that by carefully applying a selected subset of cFSM base functions the computational effort might be significantly reduced without deteriorating the results.
Two options for reducing the number of cFSM base functions have been tested: (i) ‘manual’
or user selection of a subset of the cFSM base functions, and (ii) ‘automatic’ selection of the base functions based on the eigen-value associated with Eq. (2.7). In the ‘manual’ method those cFSM functions that likely have minor importance are disregarded. Namely: higher cross-sectional L modes are disregarded, since these L modes involve smaller transverse waves, which are practically irrelevant in normal conditions. O modes rarely have high participations and therefore all O modes are disregarded. In the example this reduces the number of cross-section deformation modes to 10 (in case of a C section), independent of the cross-section discretization, thus, the total number of cFSM base functions is equal to 10mmax
where mmax is the number of various longitudinal half-waves.
The ‘automatic’ method utilizes the fact that cFSM base functions themselves are eigen-functions, i.e., buckling modes of a column problem. Those base functions corresponding to eigen-values larger than r times the minimal eigen-value are disregarded. Thus, the only necessary parameter is r. Note, this option influences both the cross-section deformation modes and the longitudinal wave-lengths, by filtering out base functions with important cross-section deformations but with unrealistic longitudinal wave-lengths (e.g., G modes with many small longitudinal waves, or L modes with very long half-waves, etc.).
Table 4.2: Change of GDLO participations and errors for the first 30 buckling modes of the beam problem with reduced number of cFSM functions:
‘manual’ reduction on the left, ‘automatic’ reduction on the right
mode DG DD DL DO D-error
nr % % % % %
min 0.4 0.0 -0.7 -8.2 0.1
max 7.6 7.9 0.1 -3.3 1.4
average 0.9 4.9 -0.3 -5.5 0.7
mode DG DD DL DO D-error
nr % % % % %
min -4.1 -3.4 -1.6 -5.0 0.3
max 5.5 3.9 4.1 4.5 3.4
average -0.5 1.7 0.4 -1.6 1.1
The GDLO participations have been calculated for the column problem of the previous Section with a reduced number of cFSM base functions. For the ‘manual’ reduction all G and D modes and the first 4 L modes have been considered (remaining L and O modes are disregarded). This reduces the number of cFSM base functions to 320 (instead of the original 2432), with mmax=32. In case of the ‘automatic’ reduction r=30 has been used, which yields to similar (but slightly smaller) number of base functions, namely: 283, while keeping some or all of the cross-section modes from all four buckling classes (thus, some O modes, too). The results are summarized in Table 4.2 (for the first 30 FE buckling modes), where the change of the participations and errors are given with respect to the results obtained with all the cFSM base functions, see Table 4.1. The minimum, maximum and average changes are given.
Both reduction methods work reasonably. There are no significant changes in the results: the increased error is typically negligible, and the GDLO participations are essentially unchanged.
The most important change is that in case of the ‘manual’ option the O contributions disappear and they are added primarily to G or D contributions (whichever is more dominant).
A properly selected subset of cFSM base functions can significantly decrease the problem size and therefore significantly decrease computation time. It might be interesting to mention that in case of the specific example the computation time on an ordinary PC (circa 2009) could be as long as 1 hour if all 2432 cFSM base functions are considered, but dropped to less than a minute if a reduced subset with 300 functions are used (using MatLab).
As far as the difference between the ‘manual’ or ‘automatic’ reduction is concerned, both options are effective and applicable. Still, the ‘automatic’ option seems to be advantageous since (i) it requires less judgment and input from the user, (ii) it might lead to more accurate participation results, (iii) it is easy to control the size of the problem and/or the accuracy of the results, and (iv) it yields a smaller number of cFSM base functions (compared to the
‘manual’ option and assuming approx. the same accuracy of the results, which should become more evident for longer members.)
4.5 Summary and continuation of the work
In this Chapter the method developed for the identification of FEM-calculated deformation modes has been presented. The method employs the cFSM base functions, the linear combination of which is used to approximate the deformations. Since in the cFSM base functions the various mode classes are separated, the separation can readily be done for the linear combination, too, which directly leads to the contribution of each mode class to the analysed displacement field. The method is illustrated for the identification of buckling modes of pinned-pinned columns. A way to reduce the number of employed cFSM base functions is also proposed and illustrated. (Thesis #3 is based on these results, see Chapter 6.)
The proposed identification method induced some further research work. First, the method has been tested for other end restraints [3/4-4/4], and it was concluded that the simple sinusoidal cFSM base functions can be applied to other-then-pinned end restraints, too, provided the transverse translational displacements are restrained.
Furthermore, Joó (with the contribution of the Author) applied the method for beam problems, including Z-shaped members with intermediate elastic restraints (as in case of e.g., purlins with trapezoidal sheeting). It was also demonstrated how the identification method can be used in design calculations, see [5/4].
Li and Schafer further extended the method. First, they investigated how other FSM longitudinal shape functions can be utilized to handle various end restraints. Finally, they proposed a so-called generalized set of cFSM base functions which was proved to be applicable to members with arbitrary end restraints. Later they also applied the identification method for deformations obtained from (geometrically and materially) nonlinear finite element analysis [6/4-8/4]. This work has been continued till lately by Li [9/4]
It is also to mention that Nedelcu has been proposed a similar mode identification method, but with using base functions distilled from GBT, see [10/4-12/4].
5 Analytical formulae for global buckling
5.1 Introduction
5.1.1 General
The birth of cFSM made it possible, and CUFSM software made it very easy to calculate critical loads for classical global buckling (e.g., flexural buckling) and compare the results to classical analytical solutions (e.g., Euler formula in case of flexural buckling). The comparison revealed some differences, the two most important ones being that (i) cFSM global buckling critical forces were a few percent higher than those from classical analytical solutions, and (ii) cFSM global buckling critical forces typically (but not always) were found to tend to a finite value as the member length approaches zero while analytical solutions predict infinitely large critical values for extremely small length.
In order to provide explanation for the experienced differences, research work started aiming to derive alternative analytical formulae for global buckling. The alternative derivations imitate the cFSM assumptions and will be referred as formulae based on shell model, while the classical analytical formulae will be referred here as ones based on beam model. Note, the term beam is used to describe a model where the beam or column member is modelled by a one-dimensional element, that is line, to which cross-section properties are assigned. On the other hand, the term shell is used to describe a model where two-dimensional elements are used, in this specific case flat strips, and both in-plane (membrane) strains/stresses and out-of-plane displacements (i.e., bending strains/stresses) are considered.
First, in Section 5.1.2, the initial assumptions are summarized, while Sections 5.2 and 5.3 presents the main steps and most important results of the derivations with neglecting or considering in-plane shear deformations, respectively, based on [1/5-5/5]. Demonstrative numerical examples are also provided.
5.1.2 Initial assumptions
An illustration of the member as well as the applied global (X,Y,Z) and local (x,y,z) coordinate systems are presented in Figure 5.1.
Figure 5.1: Coordinate-systems, basic terminology
The applied basic mechanical assumptions intend to imitate those of a FSM or shell FEM solution. More exactly, the assumption system closely follows that of the semi-analytical FSM, as implemented in CUFSM, see [16/1-18/1].
For the analysed member it is assumed that: (i) the analysed member is a column, (ii) the column is prismatic, (iii) the column is supported by two hinges at its ends, (iv) the column is loaded by a compressive force (uniformly distributed along the cross-section), (v) its material is linearly elastic, and (vi) it is free from imperfections (residual stresses, initial deformations, material inhomogenities, etc.).
As far as boundary support conditions are concerned, the applied longitudinal shape functions correspond to ‘globally and locally pinned’ and ‘free to warp’ support conditions. More precisely, for both column ends: (i) local transverse translations (i.e., in the x and z direction) are restrained, which also means that global transverse translations (i.e., in the X and Z direction) are restrained, (ii) translations in the y or Y direction can freely occur, i.e., the cross-section warping is allowed, (iii) local twisting rotations of the strips are restrained, which also means that global twisting of the cross-section is restrained, (iv) and finally local rotations about the strips’ local x-axis can freely occur, as well as global rotations about global X and Z axis can freely take place.
For the member deformation and displacements we assume that (i) the member is modelled by 2D surface elements, i.e., strips, (ii) in-plane (membrane) and out-of-plane (plate bending) deformations are allowed, (iii) for the in-plane behaviour a classical 2D stress state is considered, (iv) for the out-of-plane behaviour a classical Kirchhoff plate is considered, and (v) displacements are constrained to global buckling mode.
For the derivations the energy method is used. The total potential of the member is expressed, and critical force is searched by utilizing that in equilibrium the total potential is stationary.