## Inelastic Seismic Response of Structures

- Under relatively strong earthquakes, structures undergo inelastic deformation due to current seismic design philosophy.
- Therefore, structures should have sufficient ductility to deform beyond the yield limit.
- For understanding the ductility demand imposed by the earthquake, a study of an SDOF system in inelastic range is of great help.
- The inelastic excursion takes place when the restoring force in the spring exceeds or equal to the yield limit of the spring. or this, nonlinear time history analysis of SDOF system under earthquake is required; similarly, nonlinear analysis of MDOF system is useful for understanding non-linear behaviour of MDOF system under earthquakes.
- Nonlinear analysis is required for other reasons as well such as determination of collapse state, seismic risk analysis and so on.
- Finally, for complete understanding of the inelastic behaviour of structures, concepts of ductility and inelastic response spectrum are required.
- The above topics are discussed here.

## Non linear dynamic analysis

If structure have nonlinear terms either in inertia or in damping or in stiffness or in any form of combination of them, then the equation of motion becomes nonlinear. More common nonlinearities are stiffness and damping nonlinearities. In stiffness non linearity, two types of non linearity are encountered :

- Geometric
- Material (hysteretic type)

Figure shows non hysteric type non linearity; loading & unloading path are the same.

Equation of motion for non linear analysis takes the form

## Elasto-plastic non linearity

## Bidirectional Interaction

Bidirectional interaction assumes importance under:

- Analysis for two component earthquake
- Torsionally Coupled System

For such cases, elements undergo yielding depending upon the yield criterion used. When bidirectional interaction of forces on yielding is considered, yielding of a cross section depends on two forces. None of them individually reaches yield value; but the section may yield.

## Inelastic response spectra

## Construction of the spectra