Modal Superposition vs Response Spectrum Analysis
Modal Superposition vs Response Spectrum Analysis
(OP)
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,
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,






RE: Modal Superposition vs Response Spectrum Analysis
RE: Modal Superposition vs Response Spectrum Analysis
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.
RE: Modal Superposition vs Response Spectrum Analysis
After your clarifications and enlightments couple of question comes in mind.
[X]= normalized eigenvectors matrix;
[M]=Lumped mass matrix
[I]=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]*[I] / [X]T*[M]*[X]
Modal Displacement (MD) = [X]*[MPF]*[Sa]/(p2)
Base Shear = [K]*[MD]
RE: Modal Superposition vs Response Spectrum Analysis
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: Mi = {Ai}T[M]{Ai]}
- calculate generalized loads: Pi = -{Ai}T[M]{r}a(t)
- Use RSA/Duhamels Integral/Time integration.... to calculate the response of each mode {Ui(t)}
- Combine responses into overall behavior: {x(t)} = [A]{Ui(t)}
- Calculate equivalent static loads: {Q(t)} = [k]{x(t)}
- Base shear: V(t) = [I](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: Mi = {Ai}T[M]{Ai]}
- calculate modal participation factors: ai={Ai}T[M]{r}/Mi
- Given the natural frequencies of each mode get the maximum response acceleration from a relevant response spectra - Sa
- Base shear: Vmax = Miai2Sa
Part in bold is RSA