The idea behind the modal superposition transient stress analysis is to sum factored mode shapes (eigenvectors) from a modal analysis to calculate the structure's response under dynamic loading.
Then,
{u}=[Phi]{Modal Coordinates Vector}
where, [Phi] is the modal matrix (consisting of mode shapes) and {u} is the displacement vector.
However, this idea is used to calculate stresses based on the following expression:
{Sigma}=[Modal Stress Matrix]{Modal Coordinates Vector}
Now the question is, how modal stress matrix is defined and what's its physical interpretation?
Thanks
Hi. You may find some information on how the modal displacements are calculated in the following paper:
https://app.box.com/s/0e3zkk9e6dedhexf89byq9ci5w7zeoxm
Further,the strain is calculated from the displacment by differentiating the displacement field in the direction of the stress component ( in FEM this is accomplished by multiplying the displacement vector with he B-matrix or the strain displacement matrix to get the strain matrix). The stress matrix is obtained by multiplying the strain matrix with the D-matrix or the material matrix.
The meaning of stress and strain are the same as in a static analysis but the total values of various stress components are equal to the sum of those of the modes (just like for the displacements). In order to make sure that all stress components are adequately captured you must account for enough modes. Hope this helps.
I have never used modal stress matrix (MSM), since stress can be obtained directly from the known displacement field but I guess it must have its application.
Mentioned expression:
{Sigma}=[Modal Stress Matrix]{Modal Coordinates Vector},
is actually a mode superposition of stress measured with respect to the normalized (modal) coordinates:
[Modal Stress Matrix]=[constitutive matrix] [modal strain-displacement matrix].
Conclusively, it seems that MSM is a matrix composed of stress vectors measured with respect to the normalized (modal) coordinates.
Hope this helps a bit.
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