Relation between modal shapes and degrees of freedom and response spectrum (seismic analysis)

Degrees of freedom of a component will decide the no of modes and frequencies the component/system have. Rigid body will have only 6 DOF and hence can only have 6 modes and frequencies. Physical system/components that are flexible have infinite degree of freedom and hence can have infinite eigen values (frequency) and eigen vectors (modes). But in the FEM model, the no of degrees of freedom are "finite" and hence the eigen values and eigen vector. The no of eigen values and modes in typical meshed FEM model is = total degrees of freedom of model= no of degrees of freedoms per node*no of nodes.

What do you want to do for your second question is not clear.

Agree with the other poster that you will have the same number of mode shapes as degrees of freedom.

I don't 100% follow your second question either (in the edit). But (just guessing) the dominant/most important mode for a particular direction will be that with the highest (modal) participation factor. For many structures (especially overall buildings).....that will be among the lowest modes. (Which means the highest periods.)

bojoka said:

they each represent the response of a SDOF system, so they would each be the response to an excitation of the blue wiggly lines in the first picture?

Click to expand...

To be clear, the first picture is showing multiple mode shapes of a cantilever column that is MULTI degree of freedom. In reality, response spectrum would approximate the behavior of your structure using a SINGLE degree of freedom system with a frequency similar to the Mode 2, Mode 3, Mode 4.... frequencies. So, it always be "mode 1" of the single degree of freedom system. Just a system with a different stiffness or mass such that it's frequency changed. I tend to think of this as a series of cantilevers of different lengths (and hence difference frequencies).

And, yes, the longer period is usually the one that dominates the behavior of the structure... but not always.

Which text ? will you provide some snaps ? I think the first picture does not represent the mode shapes of MDOF system.

I preferred copy and paste some figures from some books related on this subject.



The above picture depicts mode shapes for 4 storey bldg from Seismic Design of Buildings to Eurocode 8 ( Y. Elghazouli )




And this snap from the book Seismic Design of Concrete Buildings to Eurocode 8 depicts a) three degrees of freedom model; (b) elastic and design acceleration spectra; (c) natural modes and periods; (d) storey forces and base shears in the three modes and for lateral force method, LFM (kN); (e), (f) displacements, storey drift ratios, shear forces (kN), bending
moments (kNm) from modal analysis with one or three modes and from the lateral force method.