Pushover analysis is the common name of a type of procedures that uses simplified nonlinear static analysis. A modal response spectrum analysis is a dynamic linear analysis. Modal response spectrum analysis is more suitable for problems involving the structural design of new structures, while pushover analysis is more indicated for assessing the seismic vulnerability of existing structures. There are many pushover analysis procedures. The most popular are the ATC40 capacity spectrum method (as an example, see the following paper: http://www.sciencedirect.com/science/article/pii/S1350630714003732), the displacement coefficient method (FEMA273), and the N2 method (Eurocode 8).

Thank you very much sir. what is difference between Time history method and response spectrum method. 

Of course  Modal response spectrum analysis is quiet  preferred analysis  which is  more useful for problems involving the structural design of new  constructed  structures,On the other hand e pushover analysis is more suitable in the analysis of  seismic vulnerability .

These are not methods of analysis, but academic thoughts which don't have any practical use. The only one way is the civil engineer's excellent static and dynamic knowledge and sense. 

Pushover analysis is a nonlinear static analysis to estimate seismic structural deformations. Seismic analysis normally performs as a dynamic analysis which may be linear or nonlinear. In pushover analysis the capacity of the structural element will be associated with maximum displacement.

Response spectrum is a plot of the maximum response of a series of oscillators of different natural frequencies. The spectrum can be used to find the response of any linear system based on its natural frequency .

Both methods are used in seismic analysis.

Response spectrum is using for linear elastic design for modal analysis.

while time history and pushover-analysis used for non linear inelastic design.pushover analysis based on code spectra and your model time period value. we pick the time period of our mode.e.g. first mode. and select the spectra from code. In time history we picked the data(spectra,time etc) of our own matching site and we check it for faults that are occuring near our site.

Pushover analysis is commonly used to evaluate the seismic capacity of existing structures and appears in several recent guidelines for retrofit seismic design. It can also be useful for performance-based design of new buildings that rely on ductility or redundancies to resist earthquake forces.

(RSA) is a linear-dynamic statistical analysis method which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure. RSA provides insight into dynamic behavior by measuring pseudo-spectral acceleration, velocity, or displacement as a function of structural period for a given time history and level of damping. It is practical to envelope response spectra such that a smooth curve represents the peak response for each realization of structural period.

Until now, I think the RSA is the suitable one. the pushover analysis is under study