Keywords: nonlinear simulation, structural damage, seismic design, vulnerability.

In the last 40 years, the risk to life in earthquakes has been reduced considerably. Earthquakes that resulted in thousands of fatalities well into the Twentieth Century nowadays result in only a few. This is due to intensive research in seismic structural safety. New seismic codes will further increase future seismic protection levels.

But since the seismic threat has remained roughly constant, the relative cost of building investments has shifted strongly away from the structure towards building installation and equipment, leading to a dramatic increase in material losses. In all recent California earthquakes (San Fernando 1971, Imperial Valley 1979, Loma Prieta 1989, Northridge 1994), this trend has been confirmed. The recent FEMA pre-standards guidelines were intended to reduce these losses [1], by controlling the phase from initial damage freedom to final seismic collapse using additional design limit states, characterised by suitable seismic damage indicators.

Denoted as performance-based seismic engineering, these new a-seismic design concepts put seismic damage at their conceptual centre. However, they leave several questions unanswered. For example: which damage measures are most suitable? How to apply such indicators as design aims? The present work attempts to answer these questions.

In the structural earthquake safety research of the last 30 years, many proposals for seismic damage indicators were developed, especially for a better understanding of the complicated seismic failure process of RC structures. However, all were either local or too empirical for the desired damage estimates of complete structures [3,5]. In Section 2 it becomes clear from these classical indicators that more modern damage indicators exist These describe the path of RC structures up to failure more correctly, and are constructed from physical state variables, and which range from 0.00 (undamaged initial state) to 1.00 (failure damage), as desired.

Due to the degradation of structural resistance over service-time or during a strong earthquake, seismic damage simulations are nonlinear processes. Thus, in Section 3 the manuscript explains the storage of damage information in the tangential stiffness matrix and the internal force vector  [4]. Thereby the paper simplifies the seismic action as a quasi-static process:

To justify the neglect of dynamic actions in the seismic analysis, we introduce a pushover-analysis to be solved in Section 4. One reason for this is to evade nonlinear time-history analyses, but the main reason is merely to gain statistical knowledge on the properties of future seismic events. Thus, the information on earthquake excitations in design codes (generally a design spectrum) is statistically more reliable, and leads directly to quasi-static pushover-analyses [2]. Based on natural frequencies and corresponding vibration modes of the linear-elastic structure and on the code design spectrum, maximum horizontal seismic loads are evaluated. With these substitute loads, a quasi-static nonlinear analysis is performed. This is known as the pushover-analysis [2].

In Section 5, is identified as an excellent entry for damage description , especially for seismic damage. We then derive from eigenvalues , of , the main tangential stiffness

sets of damage indicators , due to standard definitions. Those indicators range from

• for the undamaged state,
• for the failure-damaged state.

Such damage indicators can thus describe all limit states of the FEMA proposals [1].

Besides the eigenvalues of one can also use eigenfrequencies of some infinitesimally small harmonic vibrations, superposed on a nonlinear quasi-static load-deformation state of the structure. The latter delivers for zeros of the statement , identical with the failure condition .

Finally the paper proves the excellent usefulness of the proposed concept using three examples, an RC frame under vertical loads, under horizontal seismic action, and by a partly damaged RC cooling tower shell.

1

FEMA 273, "Guidelines for the Seismic Rehabilitation of Buildings", Federal Emergency Management Agency, Washington DC, USA, 1997.

2

V. Kilar, P. Fajfar, "Simplified Pushover-Analysis of Building Structures", In: Proceedings 11th World Conference on Earthquake Engineering, 59-66, Acapulco, Mexico, 1996.

3

W.B. Krätzig, K. Meskouris, "Assessment of seismic structural vulnerability as a low-cycle fatigue process", In: Ph. Bisch, P. Labbé, A.Pecker, (eds.), Proc. of the 11th European Conference on Earthquake Engineering, 161-178, A.A. Balkema, Rotterdam, 1999.

4

W.B. Krätzig, "Multi-level modeling technique for elasto-plastic structural responses", In: D.R.J. Owen, E. Onate, E. Hinton, (eds.), Computational Plasticity. Proc. 5th Int. Conf., Vol. 1, 457-468, CIMNE Barcelona, 1997.

5

I.F. Meyer, W.B. Krätzig, F. Stangenberg, "Damage prediction in reinforced concrete frames under seismic action", EEE-European Earthquake Engineering 3, 9-15, 1988.

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