Non linear time history analysis#

Computing structural response#

To compute the nonlinear response of the single-degree-of-freedom (SDOF) oscillators, the Vulnerability Modellers Toolkit (Martins et al 2021) was used. The toolkit is equipped with the open-source finite element model software OpenSees through its Python implementation (Zhu et al. 2018). The nonlinear behaviour was introduced in numerical models through the uniaxial material Pinching4 and the capacity curves previously introduced by the user. The numerical model allows for cyclic strength degradation that is implemented in three stages: (i) unloading stiffness degradation, (ii) reloading stiffness degradation and (iii) strength degradation. The default degradation implementation was adapted from simpleSDOF4.tcl by Vamvatsikos (2011) and makes use of the energy option of Pinching4


Vamvatsikos, D. (2011) Software – earthquake, steel dynamics and probability,

Martins, L., Silva, V., Crowley, H. and Cavalieri, F. (2021) Vulnerability Modellers Toolkit, An Open-source Platform for Vulnerability Analysis. Bulletin of Earthquake Engineering. DOI:

Zhu, M., McKenna, F. and Scott, M.H. (2018) OpenSeesPy: Python library for the OpenSees finite element framework. SoftwareX, 7: p. 6-11. DOI: