Journal of Engineering Mechanics ASCE, 122 (1996), 1149-1158.


Zdenek P. Bazant
Northwestern University
Evanston, Illinois 60208, U.S.A.

Milan Jirásek
Swiss Federal Institute of Technology
LSC -DGC, EPFL, 1015 Lausanne, Switzerland


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 point.)

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 multibay-multistory frame.

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 response.

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EPFL / July 1, 1996 /