Modal Superposition vs Response Spectrum Analysis

[JoeH78]

Dear Colleagues,

I wish to know what is the real difference and distinctive part between the 2(MS and RSA) techniques. Even sometimes in some articles I've observed that these 2 methods are referred to as if they are 2 different solution techniques. But actually mode superposition is a must for RSA and is preliminary step to obtain the modal displacement.

your comments will be appreciated,

Regards,

 

[amendale]

Both MS and RSA are two parts of the same method - modal analysis. Modal superposition is used to combine responses for structures having more than one degree of freedom (and thus having more than one mode). Response spectrum analysis is used to obtain the the maximum acceleration/velocity/displacement for a given mode/period. Both methods are used in modal analysis.

 

[JoshPlumSE]

Joe -

It might help to know the context in which the term "modal superposition" was used.

Response spectrum analysis generally uses multiple modal responses (weighted based on the modal participation factor) and then combines them using some statistical means to come up w/ a maximum statistical response of the structure. It is an easy method to use, especially when the exact nature of the input motion is not known.

Modal Super position is often used in a "time history" analysis as a means of reducing the analytical cost of the analysis. By that I mean to say that the amount of computer computation are reduced compared to a direct integration procedure. Essentially, you can use the structures Eigen modes as a means of obtaining the time history response of the structure to a given input motion.

 

[JoeH78]

Thank you all in advance,

After your clarifications and enlightments couple of question comes in mind.
[ul]
[li]Can RSA or MS be used individually in seismic design to obtain the base shear force (for example using the design spectra in UBC 97)? What causes me to ask this question is that, I believe that MS is proportional way of expressing natural frequency response of structure, I think that it can not be used individually in determination of absolute forces or modal displacements, instead it should be used with conjuction of other methods/datas e.g. design spectra.[/li]

[li] I think that I'm confusing the design spectra with RSA. What I inferr is, RSA can be used to derive the design spectra for different type of loading, ground conditions and dampings. Please confirm my reasoning whether it is wrong or not? [/li]


[li]It would be a good idea if you can give an pseudo-examples of how to obtain the base shear for those 2 methods? For example,
[X]= normalized eigenvectors matrix;
[M]=Lumped mass matrix
=Identity matrix
(p)= frequency obtained from modal analysis
(Sa)= Design spectra ordinate from UBC-97
[BS] = Base Shear
[K] = Stiffness Matrix
Modal participation factors (MPF) = [X]^T*[M]* / [X]^T*[M]*[X]

Modal Displacement (MD) = [X]*[MPF]*[Sa]/(p²)

Base Shear = [K]*[MD]
[/li]


[/ul]

 

[amendale]

Joe,

To answer your questions:
1) MS is just a method by which a MDOF system is uncoupled into individual modes, which allows us to calculate the response for each mode individually, and then combines the individual response back into the overall behavior. MS does not allow for a solution on its own, it must be used in conjunction with other methods such as RSA, Duhamel's integral, or time-step integration. MS is just a tool that is used to simplify analysis for MDOF systems. RSA is a method that gives you the maximum response for a given mode. If you have a SDOF system RSA can be used on its own to determine the response (base shear, acceleration...). However if you have a MDOF system then you need MS to analyze the contribution of each mode, if you want an accurate answer.

2) A single response spectra is derived from a single input acceleration record, such as a recorded earthquake. The frequency content of the input acceleration is analyzed to give maximum responses for varying natural frequencies of structures. A single response spectra is not as smooth as you see in the codes, since a single input acceleration has varying frequency content. The way a design spectrum is derived is by combining many different response spectrums for different input accelerations (that are characteristic for that location), in order to envelope the potential responses. It is then smoothed to give you a nice equation that you see in the code. Response spectrum analysis can use either the design spectrum or the exact response spectrum, but in most cases the design spectrum is used due to uncertainty and conservatism.

3) Here is brief walk-through of RSA and MS:

MS:
- define mass [M] and stiffness [K] matrices
- use eigenvalue analysis to derive natural frequencies and mode shapes: det([K] - w²[M])[A] = 0
- calculate generalized masses: M_i = {A_i}^T[M]{A_i]}
- calculate generalized loads: P_i = -{A_i}^T[M]{r}a(t)
- Use RSA/Duhamels Integral/Time integration.... to calculate the response of each mode {U_i(t)}
- Combine responses into overall behavior: {x(t)} = [A]{U_i(t)}
- Calculate equivalent static loads: {Q(t)} = [k]{x(t)}
- Base shear: V(t) = (Q(t)}

The parts in bold are part of MS, everything else is algebra, dynamics.

RSA:
- define mass [M] and stiffness [K] matrices
- use eigenvalue analysis to derive natural frequencies and mode shapes: det([K] - w^2[M])[A] = 0
- calculate generalized masses: M_i = {A_i}^T[M]{A_i]}
- calculate modal participation factors: a_i={A_i}^T[M]{r}/M_i
- Given the natural frequencies of each mode get the maximum response acceleration from a relevant response spectra - S_a
- Base shear: Vmax = M_ia_i²S_a

Part in bold is RSA