What is the magnitude of the deflections from Modal Analysis?

Question

What is the magnitude of the deflections from Modal Analysis?

Answer

The deflections produced by Modal analysis in Tekla Structural Designer are not “real”.  They are essentially dimensionless and can be of any size or scale.  For each mode, they describe the shape of the mode, but this shape could have any size (amplitude) depending on the scaling method used.

In order to derive a real amplitude and hence deflections, the energy input into a structural system needs to be specified in some manner and the excitation 'response' (in terms of real displacements and associated forces) produced by this solved for.  This would require:

1. A definition of the energy applied to the system - commonly some form of a time vs force or acceleration function - AND
2. A more complex form of analysis such as for example Response Spectrum Analysis (RSA), "Time History" or "Time step" analysis. 

While such more complex forms of analysis exist, this is NOT what Tekla Structural Designer's Modal Analysis is.  For more technical information about what it is (and is not) see the topic Modal Analysis.

So, the Modal analysis deflections have no real magnitude or unit of displacement.  Having said this, what then determines the modal deflection values reported?  See the next section.

Mode Shape Scaling

The basic theory for modal normalization, or scaling, is as follows:

Two common forms of mode shape normalization (i.e. ways of reporting) are Mass scaling and Unity scaling.  Until more recently, the shapes reported (in Tekla Structural Designer) were always normalized to Mass not unity.  Release 2022 SP4 introduced a new option to report mode shapes normalized to Mass or to Unity.  For more about this see the 2022 SP4 Release Notes.

Unity Normalization

Normalizing to unity means the largest (resultant) translational displacement of the shape is set = 1.0 and all the other nodal displacements are scaled accordingly, such that the relative shape is unchanged.  The unity scaled mode shape is commonly used to calculate Modal Mass.  For more about this see the topic Modal Analysis.

Mass Normalization

Scaling in this manner generally means the deflection values are very small numbers.  Furthermore, the larger the structure, the larger the mass and so the smaller the modal deflection values.  Scaling to mass satisfies the following equation:
 

Image

Where:

Image

  is the mode shape vector and 

Image

is the mass matrix.

Note that this is the equation for what is termed Modal Mass, which you will thus appreciate could have any value, depending on the method of modal scaling being used.  When the equation evaluates to dimensionless unity as shown above, the mode shape is said to be 'scaled to Mass' or 'Mass normalized'.

Any good reference on Vibration analysis theory will give more detail on this topic should you need it.  We can recommend the following:

• Anil K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering (4nd Edition), Pearson, 2017.