Strand7: Web notes: Nonlinear buckling

Buckling analysis - linear vs nonlinear

One of the most common questions that we get asked by our users is:

"What sorts of prob lems 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}

[K] = stiffness matrix

lcr = eigenvalue for buckling mode.

[Kg] = stress stiffness matrix. This matrix includes the effects of the membrane loads

on the stiffness of the structure. The stress stiffening matrix is assembled based on

the results of a previous linear static analysis.

{d} = displacement vector corresponding to the buckling mode shape.

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 is 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 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.

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 lateral loads 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 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 difference 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 a sufficient number of elements to adequately

capture these mode shapes. If for instance a structure is modelled from the linear Quad4

elements then there must be a sufficient number of 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 recommended 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.