Regarding design of a dam(or any similar solid monolithic structure), from Structural-Dynamics point of view, with/without computer coding(e.g. Matlab, Python, etc.); how can we compute Modal-Mass and Modal-Participation-Factors?
In addition to that, by which method can we compute the dam response when subjected to any specified horizontal ground acceleration(such as horizontal contribution of the seismic excitation)?
Dear professor Kabir Sadeghi ,
Many thanks for the outstanding and magnificent response, which I do appreciate very much.
Modal Dynamic Methods: Dynamic response analysis is typically performed using finite element modal analysis. The major modes of vibration are calculated, and the response of the structure to the earthquake is expressed as a combination of individual modal responses. There are 2 acceptable techniques for modal analysis, Response Spectrum Analysis and Time-History Analysis.
1)- Response Spectrum Method: In the response spectrum method, the modes of vibration determined from finite element modeling are amplitude weighted by a response spectrum curve which relates the maximum acceleration induced in a single degree of freedom mechanical oscillator to the oscillator's natural period. Because the timing of the peaks of individual modal responses is not taken into account, and because peaks of all modes will not occur simultaneously, modal responses are not combined algebraically. Modal responses are combined using the SRSS (square root of sum of squares) or the CQC (complete quadratic combination) methods.
2)- Time-History Method: The time-history method is a more rigorous solution technique. The response of each mode of vibration to a specific acceleration record is calculated at each point in time using the Duhamel integral. All modal responses are then added together algebraically for each time step throughout the earthquake event. While this method is more precise than the response spectrum method for a given acceleration record, its results are contingent upon the particulars of the acceleration record used. For this reason, time-history analysis should consider several accelerograms.
To compute the modal mass, modal participation factors and structure response for a solid monolithic structure like a dam, it is better consider solid element in the computer software such as ADINA, ANSYS or the like and for the formulas please refer to the vibration and seismic codes or structural dynamic books.
To APPLY THE PSEUDO DYNAMIC PRINCIPLES FOR FINITE ELEMENT ANALYSIS PLEASE FOLLOW THE FOLLOWING STEPS:
Step 1: Determine natural frequency and mode shape of the dam independent of reservoir interaction.
Step 2: Determine the effect of the reservoir on the natural frequency and the modal participation factor.
The generalized modal vibration equation is as follows:
M *(d2Z /dt2 * K (Z) = ag (Equation 1)
Where:
M*= m(uX 2 + uY 2 ), uX , uY being the modal deflection of a node in the X (horizontal) and Y (vertical) directions, and m being the mass associated with the node. The sum is over all nodes in the finite element model.
K*= Generalized stiffness. The change of total elastic strain energy with respect to a differential variation in modal amplitude.
Z= Modal amplitude, function of time only. Total nodal motion is given by Z•(uX), Z•(uY)
L = Effective driving force factor, m(uX + uY). If only horizontal ground accelerations are considered then m(uX ).
ag= Ground acceleration. Function of time. Using equation (1), it can be seen that the angular frequency of the mode N given by:
wN = SQR(K* = M * )
Any finite element program that is capable of modal analysis calculates N , M* , K*, L , but some may not output M* and L .
Step 3: Apply the pseudo dynamic loads to finite element model.
Step 4: Evaluation of results.
I hope I could answer your question.
Best of luck
Dear professor Kabir Sadeghi ,
Many thanks for the outstanding and magnificent response, which I do appreciate very much.
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