Softening-induced dynamic localization instability: Seismic damage in frames

Zdeněk P. Bažant*, Milan Jirásek

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


This paper analyzes dynamic localization of damage in structures with softening inelastic hinges and studies implications for the seismic response of reinforced concrete or steel frames of buildings or bridges. First, the theory of limit points and bifurcation of the symmetric equilibrium path due to localization of softening damage is reviewed. It is proven that, near the state of static bifurcation or near the static limit point, the primary (symmetric) path of dynamic response or periodic response temporarily develops Liapunov-type dynamic instability such that imperfections representing deviations from the primary path grow exponentially or linearly while damage in the frame localizes into fewer softening hinges. The implication for seismic loading is that the kinetic energy of the structure must be absorbed by fewer hinges, which means faster collapse. The dynamic localizations are demonstrated by exact analytical solutions of torsional rotation of the floor of a symmetric and symmetrically excited frame, and of horizontal shear excitation of a building column. Static bifurcations with localization are also demonstrated for a portal frame, a multibay frame, and a multibay-multistory frame. The widely used simplification of a structure as a single-degree-of-freedom oscillator becomes invalid after the static bifurcation state is passed.

Original languageEnglish (US)
Pages (from-to)1149-1158
Number of pages10
JournalJournal of Engineering Mechanics
Issue number12
StatePublished - Dec 1996

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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