Monday, June 27, 2005

Finite Element Analysis (3) Buckling

Buckling Analyses: Sudden Collapse

Introduction
    Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behaviour of the part near or beyond 'buckling' is not evident from the normal arguments of statics. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones.

    If a structure has one or more dimensions that are small relative to the others (slender or thin-walled), and is subject to compressive loads, then a buckling analysis may be necessary.

    From an FE analysis point of view, a buckling analysis is used to find the lowest multiplication factor for the load that will make a structure buckle. The result of such an analysis is a number of buckling load factors (BLF). The first BLF (the lowest factor) is always the one of interest. If it is less than unity, then buckling will occur due to the load being applied to the structure. The analysis is also used to find the shape of the buckled structure.


Evaluating Linear Instabilities
    From a formal point of view, buckling is an eigenvalue problem that is a function of the material & geometric stiffness matrices. Consequently, there will be a number of buckling modes and corresponding mode shapes.

    As with a frequency analysis, eigenvalue extraction may be carried out using a number of available methods, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods.

    An important note is that the eigenvalue method does not take into account of any initial imperfections in the structure and so the results rarely correspond with practical tests. Eigenvalue solutions usually over estimate the buckling load and give no information about the post-buckling state of the structure. Sudden buckling simply does not occur in the real world.

    So how should we know if a linear buckling analysis is sufficient ?? Carry out both a linear static analysis and a linear (eigenvalue) buckling analysis. If the max stress is significantly less than yield, and the buckling load factor is greater than 1.0, then buckling will probably not occur. If however the BLF is less than 1.0, then the buckling analysis will be linear provided that the max stress is far below yield. In all other cases, a non-linear buckling analysis should be carried out. If the component is critical to the safe operation of a system, full displacememnt analyses should be carried out.


Non-Linear Buckling
    A more practical approach is to carry out a large displacement analysis, where buckling can be detected by the change of displacement in the model. A large displacement problem is non-linear in nature. Geometric non-linearity arises when deformations are large enough to significantly alter the way load is applied, or load is resisted by the structure.

    The approach to a non-linear buckling solution is achieved by applying the load slowly (dividing it into a number of small loads increments). The model is assumed to behave linearly for each load increment, and the change in model shape is calculated at each increment. Stresses are updated from increment to increment, until the full applied load is reached. The solution becomes an iterative procedure rather than one of matrix factorisation alone, and consequently is computationally expensive.

    An interesting variation arises in the case of automotive applications. In the case of front end collision, the hood is expected to crumple (buckle) in order to absorb the energy of collision, as well as to save the passenger compartment. In such cases, we are not designing against, but for buckling.


Avoiding Instabilities
    Any structure is most efficient when subjected to evenly distributed tensile or compressive stress, such as occurring in cables, strings etc. Evidently, such modes of loading makes the best use of the material, and its strength. On the other hand bending (flexing) is the least efficient way of loading a structure. A high flexural stiffness of the structure means high resistance to buckling. This is true even if the load is entirely in-plane, since when buckling is imminent, the only stiffness that counts is flexural.

    Eccentricity of loading promotes buckling. Eccentricity means that the resultant load does not pass through the centroid of the load bearing cross section. It is safe to assume that in 100% of practical applications, loads are eccentric.

    When buckling occurs, symmetry of the part does not apply. There is no symmetry of the buckled shape, although both the part, and the loading may be symmetric. Correspondingly, when carring out an FE buckling investigations, it is advisable to implement a full 3D analysis of the structure under inspection.

    The non-linear stress strain behavior of the material reduces the stiffness at higher stress (load) levels, and hence elastic formulas from the handbooks tend to be highly unconservative.

    If a component is structurally slender, and is made of plastic, then the component faces buckling from three directions; from the low material stiffness, the large deflections producing eccentricity during deformation, and from the non-linearity of the material itself.

    By and large it is true that buckling usually occurs when compressive stress is present. But what is not evident that compressive stress can prevail in un-expected places. Shallow domes under internal pressure can develop local compressive stress regions, and make it vulnerable to instabilities.


Bifurcation & Snap Through Buckling
    In many systems a smooth change in a control parameter (the load) can lead to an abrupt change in the behaviour of the system. A simple example is the buckling of a rod. If a straight rod is compressed by a small load, it shrinks to some extent, but remains straight. For larger loads, however, it starts to buckle. Mathematically, the solution corresponding to a straight rod still exists, but it is unstable for the large load applied and very small transverse perturbations make the rod buckle. The transition from the unbuckled to the buckled state occurs via a bifurcation, that is, at the onset of the instability a new solution corresponding to the buckled rod comes into existence. In bifurcation buckling, there are two equilibrium solutions at the bifurcation point, the ordinary static strength of materials solution and the instable (buckling) solution.

    Snap through buckling occurs when a structure is subject to an increasing load that at some point causes the structure to undergo a gross deformation. Subseqent to this deformation, the structure regains sufficient stability to carry load, usually in a configuration that changes the structural load from being initially compressive to tensile. An example of this is a shallow dome in compression. If the load becomes too great, it buckles and snaps through so that the load is supported in tension.

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