Why are there limits for 1D element length and 2D element quality?

A member, such as a beam, may be composed of a number of analysis (or solver) elements .  These are termed 1D elements in TSD.  For example, if you have a primary beam which supports, say, two secondary beams, then it will be formed from three 1D analysis elements; each analysis element being between the start/end of the member and the secondary beam connection points.  The connection points and the ends of the primary beam will be ‘nodes’ or ‘joints’ in the analysis model.

Essentially this is true of any commercial analysis program which implements the Finite Element Method (FEM), not just Tekla’s.

If you are not familiar with this concept, you may find it useful to consult a reference on structural computer analysis. A very good reference we can highly recommend is “Modern Structural Analysis, Modelling Process and Guidance” by Iain A MacLeod, Thomas Telford 2005.

Tekla Structural Designer - again like all commercial structural analysis programs - implements the Stiffness Method for solution, which requires formation and solution of the global stiffness matrix of the model.  This matrix comprises terms for the individual stiffnesses of all (analysis) elements. You may recall that stiffness is inversely proportional to length.  Hence very short members may be very stiff  i.e. their stiffness matrix terms may very, VERY (astronomically) large numbers.

Computer matrix solutions may run into a problem called ill-conditioning, if there is a very (astronomically) large difference in the magnitudes of terms in the stiffness matrix.  For more on this see for example this Wikipdia article.  In simple terms, this is because computers process numbers using a finite, not infinite, precision.  While a computer can store a single number to a vast precision, it cannot store and manipulate  (i.e. perform mathematical operations on) millions of numbers to this precision.  Certainly not a regular modern PC (not a super-computer!) in a practical time frame.  Thus vast differentials in stiffness term magnitudes can lead to a loss in accuracy of the stiffness matrix computer solution. 

Such innaccuracies often manifest as equilibrium violation warnings for the Solver solution and/or mechanism warnings.  Where ill-conditioning is the only 'culprit', mechanisms will not be found since they are spurious and do not actually exist.  The error is in the solution itself, part of which identifies mechanisms.  When - i.e. at what 1D element length - the issue may occur cannot be known with any certainty, because it depends on the relative stiffness of all elements in the model, not just one element.  Hence the default error limit of 10mm is not exact, or "hard and fast" - it is a sensible initial value.

Ill-conditioning can also occur with poor quality 2D (shell) elements.  Poor quality elements are those which are highly distorted.  See the linked article for more discussion of this point.  Highly distorted 2D elements can also have astronomically large stiffness terms and hence may also lead to ill-conditioning errors.

So these are the reasons Tekla Structural Designer issues an error or warning for very short 1D elements and/or poor quality 2D elements, dependant upon the limits set under Home ribbon tab > Model Settings > Validation

Coming back the primary beam example - one can imagine that a very short analysis element would result if a secondary beam connected very close to, but not quite, at the start or end of the beam.  To avoid this, our recommendation would be to connect to the actual start/end of the beam  i.e. make some sensible simplifications in the model (we would also note that it is not really possible or practical in reality to connect one beam < 10mm from the end of another).   

So, quite apart from the potential for ill-conditioning discussed above, extremely short 1D elements and poor quality 2D elements are often indicative of poor or impractical modelling.  This then is another reason for warning of these circumstances.  In general we suggest any such warnings or errors are always investigated and addressed by altering/ improving the model geometry to remove them - not by reducing the warning and error limits.

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