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# Civil Engineering/classical damping-non linear analysis

Question
QUESTION: sir,
im performing non linear  time history analysis...i have read in a.k chopra which says that only classical damping(i.e damping matrix diagonal)is applicable in linear systems and time history can be solved through classical modal analysis.. and non classical damping (i.e damping matrix non diagonal) can be solved only in direct integration methods...i have also read that if external energy disspation devices, base isolators are incorporated in the model then the damping matrix might become non classical....so according to this concept non classical damping has to be solved only in direct integration methods irrespective of the system being linear or non linear..now my query is sap 2000 allows us to perform non linear modal time history analysis...pls enlighten me with the following doubts..
1.)modal time history analysis is nothing but the classical modal analysis...then how come sap 2000 says to perform non linear modal time history analysis..when we know that classical modal analysis is applicable only for linear systems..?
2.) Csi knowledge base of sap 2000 says it performs non linear modal time history analysis and consider the system as non linear when there are boundary elements in the system like (link elements, isolators ,dampers, springs etc)...so after reading the above statements of knowledge base for the time being with half heartedness i agree it does perform non linear time history analysis when there are boundary elements ,link elements etc are involved..so then when i introduce energy dissipation devices,dampers etc..the damping matrix will (or might) become non classical(i.e non diagonal)..then how come modal analysis is applicable for this system which has non classical damping because A.k chopra says only classical damping is allowed for classical modal analysis...
3.)can u tell me how non linear modal time history analysis is a method for performing non linear time history when it does not consider the most important non linearity which is the material non linearity...?

The damping matrix is not diagonal, but it can be diagonaled using modal analysis method. In the case of non proportional damping where damping coefficients cant be expressed as a linear combination of mass and stiffness, diagonality of damping matrix cant be achieved. It is very clear that in the case of nonlinear behavior of the structure or in the case of damper usage , one cant derive a linear relationship between damping, mass and stiffness. But how can we use modal method to resolve non Reighly damping (non proportional damping ) case? Simply it is an approximation which can be used where you can accept a near proportional damping instead of a complicated model of damping. I such a case you must check your answers and analysis to get persuaded of a good approximation.

Regards
BEH

---------- FOLLOW-UP ----------

QUESTION: sir,
i know that matrix can be diagonaled using modal analysis method.....and im using rayleigh damping only ...if im performing non linear time history analysis using modal method..then are you trying to say that even though the damping matrix doesnt become diagonal after the introduction of energy dissipation devices(i.e damping matrix is still non classical)....it still can be incorporated in sap 2000 when perform non linear modal time history analysis.....?

Dear Shoaib

I tried to say that using modal analysis in nonlinearity and non proportionality case must be regarded as an approximation which can tend to be erroneous. The significance of the error depends on amount of lumped damping. I usually use parallel methods ( like incremental method)  to check my results.

Dont hesitate if you had further question.

Regards
BEH

Civil Engineering

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