On the capacity side, three main components of uncertainty in the seismic capacity of the structure are:
Probabilistic Analysis Approaches
When defining limit states, the uncertainty detailed above is usually handled by a probabilistic analysis. In general, one would want to state the problem as w% chance of exceeding the performance goal in 'y' years (life of structure). However, this is a complex, computationally intensive reliability problem which must be solved rigorously as such.
Currently, it is more common to approach the problem as x% chance of exceeding the performance goal for an earthquake with a z% probability of occurence in 'y' years. This allows the analyst to treat the ground motion and the structure separately. There are several ways of doing this. A probabilistic response spectrum can be coupled with a deterministic "conservative" design. Or, calibrated load and resistance factors can be developed using reliability analysis or Monte Carlo simulation to give an appropriate overall reliability.
Geotechnical engineers and seismologists (as well as some structural engineers) are able to and do regularly develop estimates of peak ground motion parameters (acceleration, velocity, etc.), elastic response spectra, and even time histories corresponding to the earthquakes with z% probability of occurence in 'y' years. How this is done is discussed in the Earthquake Definition and Elastic Response sections of the notes.
The three-tier limit state format is the one most commonly found in the literature for earthquake resistant structures. This way of defining limit states can be summarized as follows:
The serviceability limit state is most certainly the most well-defined of the three limit states, as it mandates that the structure remain elastic and allows for very limited damage. The performance criteria such as drift are most often determined by the nonstructural elements such as doors and partitions. Often, this limit state will control the stiffness of the structure as the deformation capacity of nonstructural elements, and thus the allowable drifts, are quite small. This will be discussed later in the course in the context of preliminary design and member sizing.
Note that the probability of the seismic event can vary quite a bit from document to document, and that the damageability limit state is not always present. The purpose of the damagability limit state is to provide criteria addressing repairable damage somewhere in between the serviceability and ultimate limit states. Repairability of damage is increasingly an issue, as many owners would like to economically repair buildings which have only moderate damage rather than knocking them down and starting over. However, the damageability limit state raises questions. What is repairable damage? How should structures be designed so they can be repaired? What are the relative repair costs?
ultimate or safety limit state is concerned with the structure's response
to a major earthquake, usually the 'maximum considered earthquake' or
the 'maximum capable earthquake'. There is the question of how rare this
earthquake should be - 10%, 5%, or 2% in 50 years? The size of the earthquake
is handled differently in different documents.