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Time history analysis of earthquakes for structures, is a NON-LINEAR dynamic method correct? Then why is there also something called "LINEAR response history analysis"? What is the difference of this from non linear time history analysis?

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A linear analysis is ... linear. If you increase the loads by a factor of two, the results increase by a factor of two. If you increase the loads by a factor of 1 million, the results increase by a factor of 1 million, even though the results are likely to be wrong. The analysis does not understand that other effects occur when the results are so large. (Those other effects are the stiffness of the structure changes, and the direction of the load changes, and so on.)

A nonlinear analysis does not have a linear relationship between the load and result. Doubling the load may only give results 1.5 times as much (the structure stiffens) or a result 3 times more (the material yields or gets weaker for example).

Time history indicates it is a dynamic analysis. The loafs and results can change over time.

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The linear time history analysis of the response of the structure is calculated at small time intervals sequenced with appropriate seismic load, maybe even after the ground motion stops until the structure stops from vibrating.

The FEA's programs either use Duhamel integral or numerical methods to calculate stress and deformations. During the loading and unloading, the structure's stresses don't go beyond elastic linear levers.

The nonlinear time history analysis or Push-over method allows during the short sequence of time intervals again, for stresses and deformations of the structure to go beyond the elastic limit into the plastic limit with high angry absorbing deformations.

This analysis is mostly used to verify the viability of the structure in a very large earthquake to while deforming beyond a repairable state, still, give the occupants time to safely exit the structure.

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