My textbook says for mode combination, the peak value $r$ of each response quantity is determined by combining the peak values $r_n$ of the modal contributions to the same response quantity.

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From what I understand, this means that we have to calculate the response (be it displacement, the difference in displacement, the forces or moment) for each mode first, before we apply the modal combination method such as SQSS, CQC and ABS in order to obtain the total response.

This is also verified by the numerical example in the textbook.

However, in the textbook example, all the members are 1-element members, easy to interpret the above clause in the textbook. What about members (beams, columns) that are of more than one element? In FEM we often create a single member out of multiple elements.

Do we combine the element response into member response on per mode basis first, before we combine all the member modal responses into total member response?

Or do we combine the total modal response for each element, and then only combining the element response into member response?

In other words, is the modal combination done at element level or at structural member level?


1 Answer 1


You are correct that the response for each mode must be calculated before applying some modal combination methodology.

Instead of thinking about the structure as being discretized into numerous elements, I like to think of it in terms of the lumped masses and how many DOFs there are. A more complex structure will simply have more DOFs (and therefore more modes) than a simple 'lollipop' model. But you still come up with modes for the structure as a whole - not modes for individual elements within the structure.

Since you mention RSA, I'm assuming you're approaching all this from the perspective of structures subjected to earthquakes. During an earthquake, multiple modes will be excited, depending on the frequency content of the earthquake. How then, do you determine the overall structure response? RSA is a simplified tool for seismic design -- allowing you to first determine the peak response of the individual modes and then combine those responses using SRSS, CQC, etc. This is opposed to running time-history analyses (very computationally intensive, but sometimes done for complex and/or important structures).

  • $\begingroup$ I think your response is not clear with regards to my question: let's say I have a beam member with 5 elements. Do I 1) get the individual mode response for each element, get the individual mode response for the beam member, and combining the modes lastly, or do I 2) et the individual mode response for each element, combine the modes for each element, and only finally combine all of the element response into the beam member? $\endgroup$
    – Graviton
    Oct 28, 2016 at 8:34
  • $\begingroup$ @Graviton I think I might have an easier time providing a clear answer if I knew what you're ultimately trying to calculate. When you say "individual mode response for each element" what do you mean? In my experience, when working in modal coordinates, you're dealing with the matrix for the entire structure, and solving the equation of motion for the entire structure. $\endgroup$
    – CableStay
    Oct 31, 2016 at 17:17
  • $\begingroup$ "individual mode response" can mean the element displacement, or element force ( by multiplying the local stiffness with the displacement).. or anything. Say my beam has 5 elements, my understanding is that I will first obtain the element displacement for every mode, then obtain the element force for every mode, then I will get my member beam force for every mode ( by combining the elements together, and use average value at node at element joints for example), and then only use mode combination to obtain the total member beam force. Is that correct? $\endgroup$
    – Graviton
    Oct 31, 2016 at 23:20

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