What is 2nd Order Buckling Analysis? How and when would I use it?

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Background

For the full context of when you would usually use 2nd Order Buckling Analysis, see the Related Article I have analysis Warnings and/ or Errors about buckling. Why is this happening and what do I do about it?

Buckling is a regular, everyday phenomenon with which everyone - including non-engineers - will be familiar, as illustrated in the picture below of a buckled plastic ruler.  Depending on the analysis type being used, buckling can also occur in computer analysis, just as in reality, if members are insufficiently stiff to resist the compressive forces applied to them.  Note particulalry the use of the term 'stiff' here - buckling (certainly the analytical form) is a function of stiffness NOT strength.

In TSD, buckling can occur when running 2nd Order analysis (both linear and non-linear).  To understand why and for more about 2nd Order Analysis see the related article What are the "2nd Order" Analysis options on the Analyze Ribbon in TSD?.

When buckling issues occur during analysis, there will be warnings and/ or errors issued in the Solver Status tree, as shown in the picture below.  Our advice is: never ignore these!  They will state “Structure may be close to or beyond buckling limit” or “One or more structural elements buckling” together with a tooltip “Run 2nd Order Buckling Analysis”.  2nd Order Buckling Analysis can be used to diagnose the problem - to find how and where buckling is occuring.

 

• Tip - note that the form of buckling under discussion here for 1D elements (i.e. beams, columns etc) is compression - or ‘strut’- buckling NOT lateral torsional buckling (LTB) due to bending.  LTB behaviour is not described in the 1D element stiffness matrix, so does not occur for them (individually) in analysis.  So buckling of 1D elements only results from axial compressive loading, not bending.  Buckling of 2D element meshes (used in meshed concrete walls for example) can also occur and could be of any form depending on the structural form and loading.

As Euler buckling theory tells us, when the elastic critical buckling load of a member (or indeed an entire structure) is reached, it will fail in a buckling ‘mode’ or shape which is unbounded. I.e. the element (or structure) can displace indefinitely and hence has zero stiffness.  The consequence of this in terms of a numerical computer solution, is that the stiffness terms of buckled element(s) become zero which in turn can render part or all of a structure unstable.  The severity of the impact on analysis depends on many factors and can range from warnings to complete failure of solution.  In the latter case, there is no choice but to resolve the buckling issues in order to achieve solution.  For buckling warnings, our general advice is that they are never ignored, as the solution may be unreliable.

Forms and Properties of Buckling

There are essentially two forms of buckling (which may occur in TSD Analysis):

• Strut buckling - an individual member buckling on its own.
• Sway buckling - an arrangement of members, or even an entire structure, buckling laterally en masse.

Next, there are two quantitative properties which describe the buckling behaviour, these being the mode Shape and mode Factor.

Buckling Mode Shape

From elastic theory; for a single member laterally fixed and pinned at both ends (as in the picture of a ruler above), the first and lowest mode shape is a half sine wave.  Some key points are:

• The shape can have any magnitude - as mentioned above, from elastic theory it is unbounded.  So the mode shape deflections are not ‘real’.
• There will not be just one mode - again theoretically, a single member has an infinite number of ‘higher’ modes, each formed of larger multiples of sine waves, and occurring at progressively higher compression.
• Practically, in the computerized numerical solution, the number of modes is limited by the number of 1D analysis elements that structural members are comprised of - this is termed the level of discretization of the analysis model.
• Both the mode shape and factor (both inherent in 2nd order analysis and that given by Buckling analysis) will be approximate, not exact. 
• The level of approximation is determined by the number of 1D analysis elements used for a member.  At least two 1D elements (i.e. one internal node) are required for reasonable accuracy of the first, simplest and lowest mode.  The same is true of 2nd order analysis and reasonable accuracy for 2nd order effects,  which is the reason why the TSD solver model uses two 1D elements per structural member.
• From the foregoing you will appreciate that a structure comprising many members can have a huge number of modes.
  • However, we are generally only concerned about the lowest and simplest modes, since these occur at the lowest load.

Buckling Mode Factor

This is the value of load at which a particular buckling mode occurs, reported as a proportion of any given applied load.  When the factor is < 1.0, this means that buckling occurs before that full load can be applied.  So, if you tried to run 2nd Order analysis for this load, buckling will occur during the analysis and it could fail to solve. 

 

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Running Buckling Analysis

Buckling analysis is a ‘modal’ analysis.  As such it is highly analogous to Vibration Analysis, which is also a modal analysis.  Just like Vibration Analysis, Buckling Analysis gives Mode Shapes (as discussed above) which can be graphically represented to show the behaviour.  Each mode will have its associated buckling factor (again as discussed above) which is the factor of the load case/ combination you have subjected to analysis at which the mode occurs.  These are the only results that Buckling Analysis yields: Mode shapes and their factors.

So a much better idea of the location and nature of buckling can be found by running 2nd Order Buckling Analysis directly (as the Analysis Error/Warning tooltip suggests) via the Analyze ribbon as shown in the picture below.

There are some settings for this analysis type which are accessed via Analyze > Settings > 2nd Order Buckling.  In most cases these can be left as the default values.

Please note the following:

• Do not attempt to run buckling analysis for all cases and combinations!  Consider:
  • Buckling analysis is an iterative process and so - depending on the model size - can take some time to complete.
  • If buckling occurs for a load case, then it will also occur for all combinations which include that case (with a factor ≥ 1.0).  Hence is is usually a waste of time running buckling analysis for such combinations.
  • Remember - the point of running buckling analysis in this context is to find and resolve buckling issues so that 2nd order static analysis will solve with no errors or warnings, and design can then be accomplished.  So it is just a means to this end. 
  • So logically you need only to run it for the minimum number of cases/ combinations required to diagnose and resolve buckling issues.
• Buckling analysis is not a design ‘check’ as such, or something that is required regularly:
  • For buildings, the Sway/ Drift checks are the mandatory checks for overall structure stability that are required and that are automatically performed by TSD.  For more on these see the help and TUA articles The sway check, The drift check and  Why does the Status tree Report a Design > Sway Warning?.
• In some less common circumstances Buckling Analysis might also be run as a form of check e.g. to find the analytical elastic critical buckling factor for a structure or part of a structure which is beyond the scope of design standard sway/drift and design checks.  Further discussion of this point is beyond the scope of this article.

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Buckling Analysis Results - Viewing Buckling Modes

When buckling analysis is complete, only the two results discussed above are available; mode shapes and their associated factors.  To view these:

1. Just as for any other results, select the case/ combination you wish to view results for.
2. Activate Total Displacements (usually these will already be active) for both 1D and 2D elements (the latter if you have 2D element meshes subject to significant compression forces that might be buckling).
3. Select a Mode from the drop-list on the Ribbon and the buckled mode shape (at one extreme of amplitude) will then be displayed.
4. For the best appreciation of the mode Animate the shape (right-click > Animate… as shown below).
5. Remember: generally we are only interested in mode shapes with factors < 1, since these will be the ones causing analysis issues.

Selecting Modes and Animating Shapes

The brief video below shows how to do this (note that you can change the selected mode shape and case/ combination while Animation is active)

Sway buckling mode

The brief video below shows an example of this.

Buckling of 2D element mesh

2D (FE) element meshes can buckle as well - the brief video below shows an example of this.

What to do about Buckling Problems

See the Related Article I have analysis Warnings and/ or Errors about buckling. Why is this happening and what do I do about it?