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3 Things you Should Know About Defining Mass for Natural Frequencies and Response Factors
Posted on in Dynamics

Lately, we’ve been getting more and more engineers interested in the dynamic behaviour of their structures. People want to make sure the natural frequencies of their bridges aren’t too low and the response factors of their floors or staircases aren’t too high. One of the key engineering decisions in checking these values is the assignment of appropriate mass to the structure for the modal analysis, read on if you want a deeper understanding of what this decision is all about!

The first thing to understand is that dynamic analysis has to account for the mass whereas a static analysis does not. The mass of the structure is completely irrelevant when all you want to work out is the final displacement (and ultimately stresses) of your model under a load, but in dynamic analysis we need to know about mass because we are interested in how the structure moves under a changing load. Even when you just want to check the natural frequencies of your structure, what you’re really looking at is the different components of movement the structure will experience under a changing load. So we know that for a dynamic analysis we need to assign mass to the structure, but one question that comes up a lot is how much mass should we be including?

How Much Mass?

If you are interested in floor or staircase vibration response factors, you may be familiar with the SCI P354 Corus handbook. Sections 6.1.1 and 4.1.2 of this document discuss the mass that should be included in the modal analysis. These sections emphasize the importance of only including unfactored loads which can reasonable be expected to be permanent in your structure.

It is important to understand that (unlike in static analysis) increasing the amount of mass in your model is actually not conservative for calculating response factors (and other dynamic responses)! It also is not necessarily conservative to reduce the mass in your structure, because this will lead to different natural frequencies that may or may not resonate with the applied vibrations. Basically, the safest option is to include the unfactored weight of the structure and unfactored permanent loads. This applies to all forms of dynamic analysis, not just response factor analysis…

Note: If you are simply checking that your structure does not have natural frequencies below a certain threshold, then adding more mass to the structure than you expect it to have in real life may be conservative (adding mass without changing stiffness lowers natural frequencies). However, even for this application I would recommend using the unfactored mass and permanent loads of your structure because this will give the most accurate natural frequencies…

Note: The methods described below are relevant to MasterSeries Patch 2013.13.30 onwards.

The Easy Way to Add Mass

MasterSeries offers an extremely flexible way to add mass to your model for dynamic analysis. The easiest and most basic approach is to check the box for including density and add a factor of 1.0 to the chosen density load group. This will then build a mass matrix for your model based on the self-weight of the structure.

Add Mass

In the Edit Mass Load Contributions Settings: First, enable the density for self weight option and choose which load group to send the density to. Then add a factor of 1.0 to these load groups.

The material self-weight may not account for all the mass of your structure, so MasterSeries allows the user to add factors to nodal, member or area loads that you have set up in your static model. This leads us on to the alternative way to define your structure’s mass…

The Hard Way to Add Mass

In a few cases it may be preferable to the engineer to define their structural mass completely manually, without automatically including material density as described above. To do this the user must set up the appropriate loading in MasterFrame, assign the loads to load-groups and then in the dynamic add-on assign a factor to these load-groups.

If you are considering this more advanced approach, there are a few things to bear in mind:

  • There has to be at least some mass assigned to every free degree of freedom in the structure, otherwise you will get an error at the start of the dynamic analysis stating that the mass or stiffness matrix is not positive definite.
  • Nodal loads do not assign mass to rotational degrees of freedom, this means that you can’t define your mass using just Nodal Loads, you must use member loading or area loading also.
  • Masterseries will only take vertical loads into account as mass. This raises the question: how do you assign vertical loads to columns without using nodal loading? For columns (or any other inclined members) apply the member load “UDLY“, then change the “Y” to a “W”. This will distribute the vertical load over the full length of the member instead of just its horizontal projection. The same effect can be achieved using the “Density” load but you will be working with kN/m3 instead of kN/m.

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