Limit States Design
Current Approaches Used in Earthquake Engineering

An LRFD format is often utilized in the current approaches to performance-based design due to its familiarity. On the load, or demand, side the uncertainty is often divided into several parts:

• Ground motion - randomness and uncertainty treated using probabilistically based response spectrum and load factors corresponding to geographical location and soil conditions
• Structural response - even for a family of ground motions with 'similar' characteristics, structural response will have large variations
• Analysis method:
- elastic static (ESP)
- elastic dynamic (EDP)
- nonlinear static (NSP)
- nonlinear dynamic (NDP)
• Modeling - variations in mechanical and dynamic characteristics will make these uncertainties in response demand larger

On the capacity side, three main components of uncertainty in the seismic capacity of the structure are:

• Element level effects - stress, plastic hinge rotations
• Global behavior - drifts, static and dynamic instability
• Brittle failure modes - premature column fracturing or buckling

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.

Common Limit States

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:

 Limit State Objectives Performance Criteria Probability of Seismic Event Probability of Exceeding Performance Criteria Functional Damage-related Serviceability Continued occupancy of structure w/o significant interruption of service. No significant damage to structure, nonstructural elements, or contents. Stresses less than yield (elastic design). Displacements & drifts less than damage threshold. Accelerations limited to minimize falling hazards. x1% in y1 years (50% in 50 yr) u1% in v1 years (0.1% in 30 yr) Damageability Continued viability of structure with some interruption of service. Limited damage to nonstructural elements or contents and no significant damage to structure. Structure repairable. Stresses slightly above elastic limit. Displacements and drifts slightly bigger than damage threshold. Accelerations limited to minimize falling hazards. x2% in y2 years (20% in 50 yr) u2% in v2 years (0.2% in 30 yr) Ultimate (Safety) Life safety of occupants. Maintain economic-operational viability of user or owner. Life safety of occupants. No collapse or significant falling hazard. Limit structural damage to that which can be repaired economically. Plastic deformation mechanism with plastic rotations less than rotation capacity. Limit drifts and displacements to avoid instability. Limit accelerations to avoid falling hazards or attach components to mitigate potential for falling hazards. Limit rotations & displacements to values that can be repaired. Locate damage in regions that can be repaired safely. x3% in y3 years (5% in 50 yr) u3% in v3 years (0.3% in 30 yr)

Serviceability

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.

Damageability

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

The 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.

Note that institutional viability is a concern, as well as occupant life safety, especially as the economic toll of business interruption becomes greater. There is a large gap between maintaining operational and economic viability and only providing for life safety. Either another limit state may be needed, or possibly a different approach should be used.