Journal of Engineering Mechanics ASCE, 122 (1996), 1149-1158.
SOFTENING-INDUCED DYNAMIC LOCALIZATION INSTABILITY:
Zdenek P. Bazant
SEISMIC DAMAGE IN FRAMES
Evanston, Illinois 60208, U.S.A.
Swiss Federal Institute of Technology
The paper analyzes dynamic localization of damage in structures with
softening inelastic hinges and studies implications for seismic response of
reinforced concrete or steel frames of buildings or bridges. First the
theory of limit points and bifurcation of 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 adsorbed by fewer hinges, which means faster collapse.
The dynamic localizations are demonstrated by exact analytical solutions of
torsional rotation of a 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.
Summary and Conclusions
Under monotonic static loading, softening damage in frame structures
causes bifurcations of the equilibrium path in which the symmetry of
response breaks down and the damage localizes into fewer hinges which
soften faster. Although the dynamic response of a structure with softening
damage (and with fixed parameters) cannot exhibit bifurcation in time, it
becomes (for a limited time, temporarily) dynamically unstable (in the
sense of Liapunov) after the static bifurcation state has been passed.
Small imperfections cause deviations from the primary (symmetric) response
mode to grow exponentially (at least temporarily), causing dynamic
localization of damage into fewer softening hinges. Because the same
kinetic energy must be absorbed by fewer inelastic hinges, the collapse
then progresses faster.
An exponentially growing deviation from a periodic solution, causing a
similar behavior, also occurs in the dynamic response near the static limit
point. (The bifurcation point may but need not coincide with the limit
The softening-induced (time-limited) dynamic instability due to
formation of softening hinges may have serious implications for seismic
resistance of building frames and bridges. The inelastic hinges in
reinforced concrete columns or prestressed beams exhibit post-peak
softening, caused by compression failure of concrete. Steel beams can also
exhibit post-peak softening, caused by elasto-plastic buckling of flanges
and webs or by growth of cracks during earthquake. Such behavior generally
leads to the aforementioned type of damage localization.
Localizations of damage into fewer softening hinges which lead to
exponentially growing deviation from a symmetric or periodic solution have
been demonstrated for the following typical examples: Torsional rotation of
a building floor, horizontal shear of a building column, shear loading of a
portal frame, shear loading of a multibay frame, and shear loading of a
Dynamic localizations due to softening hinges cannot be captured by
analyzing the structure as a single-degree-of-freedom oscillator because
the deviation from symmetric response represents a very different mode of
Please send me an email
if you wish to receive the complete paper.
July 1, 1996 /