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One of the most common questions that we get asked by our users is:
"What sorts of problems are suited to a linear buckling analysis and when should I be considering the use of a full nonlinear buckling analysis ? What are the advantages and disadvantages of the two different types of buckling analysis ?"
Before we answer this question it is appropriate to discuss briefly the two different types of buckling analysis, how they work and how the buckling loads are obtained.
Linear Buckling
A linear buckling analysis is an eigenvalue problem and is
formulated as follows:
([K] + lcr [Kg]){d} = {0}
The eigenvalue solution uses an iterative algorithm that extracts firstly the eigenvalues (lcr) and secondly the displacements that define the corresponding mode shape ({d}). One set of these is extracted for each of the buckling modes of the structure (up to the user specified limit). Note that the displacements given by the solution are not real displacements in the normal sense. They are simply a set of scaled values used solely for display purposes.
The eigenvalue represents the ratio between the applied loads and the buckling loads. This can be expressed as follows:
lcr = Buckling Load / Applied Load.
Thus the eigenvalue is, in effect, a safety factor for the structure against buckling. An eigenvalue less than 1.0 indicates that a structure has buckled under the applied loads. Conversely an eigenvalue greater than 1.0 indicates that the structure will not buckle.
The important thing to note about this formulation is that only the membrane component of the loads in the structure is used to determine the buckling load (since the formulation of [Kg] is based solely on the membrane loads). This means that the effect of prebuckling rotations due to moments are ignored. As we shall see this has some very important implications when choosing the type of buckling analysis to use for a particular problem.
Nonlinear Buckling
A nonlinear buckling analysis can be carried out using the
standard geometric nonlinear solver. The load is applied
incrementally, from zero up towards the maximum. Normally each of
these load increments will converge in a small number of
iterations. The onset of buckling is generally indicated by
failure to converge for a particular load step. If the results
are examined for those increments just before the increment that
failed to converge, the buckling mode shape can usually be seen
to form. The load increment that failed to converge will
generally correspond to complete collapse of the structure; at
this point the stiffness of the structure is suddenly reduced and
is no longer sufficient to maintain equilibrium with the applied
loads - hence the failure to converge. In order to capture this
buckling behaviour it will be necessary to use small load steps
for the last part of the loading up to failure.
However not all structures collapse when they buckle. Many structures exhibit stable buckling modes where the loads in the structure redistribute themselves as a result of some initial buckling failure; the resulting structure is sufficient to carry these redistributed loads and thus the structure does not collapse. The nonlinear solver is ideally suited to modelling this sort of post buckling behaviour. In this case the load is stepped up past the initial buckling into the postbuckling range.
In the nonlinear analysis the stiffness matrix is updated periodically (for every iteration of every load increment) based on the current deformed shape of the structure. This is important from a buckling point of view since the effect of the pre and post buckling deformations are included in the analysis. When we talk about 'prebuckling deformations' we are generally referring to those deflections caused by the moments in the structure prior to buckling. 'Post buckling deformations' refers to those deflections that result from some initial buckling failure of the structure. The effect of this updating of the stiffness matrix will be to allow the bending and membrane stresses in the structure to redistribute themselves to reflect the current deformed geometry.
The advantage of this approach is that the nonlinear solver can be used to model postbuckling behaviour of structures, particularly when the transition through the buckling load is smooth, with no snap through. If the structure snaps from the prebuckled geometry into the postbucked shape then in many cases the nonlinear solver will have trouble tracking this rapid change in geometry and may not converge. In some situations convergence can be helped by reducing the load step size. Without special algorithms, snap through problems generally must be run using displacement control of the solution - not load control. (see the Snap Through of an Arch in G+D News 10).
The other important point that should be made about the nonlinear solver is that material nonlinearities (yielding) can be considered in addition to the geometric effects. This allows the prediction of elasto-plastic buckling modes and local crippling of sections. An example of this is shown in the following figure:
In this analysis the local buckling of a roll formed steel post was predicted, with good accuracy. Both geometric and material nonlinear effects were considered. Many postbuckling problems will involve yielding.
One of the problems with using the nonlinear solver for the solution of buckling problems is that some models will not buckle and may need special treatment. If we consider a simple euler column, that is purely in compression (i.e. only membrane loads) with no bending then this will continue to sustain as much axial load as we apply. This loading condition cannot generate any lateral load to initiate buckling. If the nonlinear solver is used for the analysis of a structure such as this then we must apply a small lateral load to help initiate the buckle. In the case of an euler column a lateral load of approximately 0.5% of the axial force for buckling would be required.
Many of the problems that fall into the nonlinear class will not have a 'snap' type of buckling failure - that is they will fail more by a large deflection type of elastic collapse. This is generally typical of any structure where bending dominates the behaviour of the structure. Consider for example a column with an axial load as well as a lateral load. The lateral load will cause sideways deflection and bending to be developed in the column as soon as the load is applied. The failure mode will ultimately be by sideways bending of the column - when the vertical force is large enough to maintain a lateral displacement in the absence of any lateral force.
The column will not have a critical load at which point it suddenly collapses; provided that the material does not yield, failure will be progessive and predicatable.
Let us now return to the question posed above. The main point that was made above was the difference between the two methods in their handling of the prebuckling deformations of the structure. Herein lies the answer to the question.
The linear buckling analysis method only considers the membrane loads in the structure and thus its use should be restricted to structures where the loads are essentially all membrane. Examples of this are:
Euler
Column - The column carries the load in pure
compression. There are no moments present in the beam.
Hoppers - Conical
hoppers fabricated from thin plate, loaded by the weight
of the material inside them, are generally loaded
principally by membrane loads and are thus suited to
analysis by the linear solver. Generally these are
fabricated from thin sheet that cannot sustain any
substantial bending moments.
The nonlinear solver can be used for most general buckling problems however it is best suited to problems where bending is an important part of the structural behaviour. In such structures the solver will generally find the buckling mode without the addition of extra loads to induce buckling. Examples of structures that should be solved using the nonlinear solver are:
Beam
structures where the beams are subjected to
UDL's, and moments due to eccentricities in the
structural geometry and loads.
Plate/Shell problems
were there is significant bending in the plates such as
structures with flat panels and a normal pressure
loading. In cases such as this the membrane loads will
exhibit a significant redistribution when the problem is
run nonlinear compared with a linear analysis.
As an illustration of the different applications of the buckling solvers consider the following problem:
Vertical Column
Consider a simple vertical column. The column is loaded in compression with varying amounts of offset.
The table below shows the critical buckling load for the linear solution with three different amounts of offset. The linear solution always gives the same buckling load, irrespective of the eccentricity of the loads, which is clearly incorrect. The calculated buckling load always corresponds to the euler buckling load for a column without eccentricity. This illustrates the point made above about the linear solver ignoring the bending loads. The solution is based on the membrane loads which in this case are always constant. It is the bending moment that increases as the offset is increased. Also included in the table is a nonlinear solution for zero offset. In this solution a small lateral load of 1N was required to provide a small lateral deflection so that buckling would initiate. The onset of buckling in this solution was indicated by non convergence of the 5300 N load increment and large increases in the lateral deflections. Agreement between the linear and nonlinear solutions for zero offset is good. The nonlinear solutions with offset do not have a distinct buckling load.
| Linear Pcr | Offset (m) | Nonlinear Pcr |
|---|---|---|
| 5310.46 | 0.0 | 5300.00 |
| 5310.45 | 0.05 | N/A |
| 5310.43 | 0.1 | N/A |
| 5310.40 | 0.2 | N/A |
The results for the nonlinear solution with increasing offset are shown in the following graph. In this case the lateral deflection at the centre of the column is plotted as a function of applied load. The linear buckling solution is also included for reference.
Note that in this simple example the structure supports postbuckling loads in excess of the euler load. This is due to the fact that the material was assumed to be linearly elastic.
The nonlinear solution has been checked by comparison with a theoretical solution for the lateral deflection of an eccentric column from 'Mechanics of Solids' by Hall. For an applied load of 4000 N with E=0.1m the error in the finite element solution is approximately 3%.
With any buckling analysis is important to use a fine mesh (as is also the case with natural frequency analysis). Since some of the buckling modes can be quite complex, particularly for the higher modes, it is necessary to have sufficient elements to adequately capture these mode shapes. If for instance a structure is modelled from the linear QUAD elements then there must be sufficient elements to provide a piecewise linear approximation to the modes of interest.
Perfect / Imperfect Structures - Typically buckling in real structures is influenced by the imperfect nature of the structure and thus a certain amount of scatter is to be expected in the results. A finite element buckling analysis in general will yield the buckling results for a perfect structure and thus will represent the upper bound of the buckling load for any structure. The expectation is that the real structure will, in practice, buckle at a load below that predicted by the finite element analysis. The effect of any imperfections will depend on the actual geometry and loading of the structure etc. If the effects of imperfections are thought to be significant then it is recommend that some attempt be made to include the imperfections into the model. In any case it is common practice to allow large safety factors when predicting the buckling strength of a structure.
For more information please contact us by
e-mail: hsh@iperv.it
