# LRFD for Geotech – Note 1

## 1.0 Concept of Limit States in the AASHTO-LRFD Framework

The AASHTO-LRFD approach uses reliability (probability) theory to quantify the uncertainty in loads, Q, and resistances, R.    In the AASHTO-LRFD framework, once the load factors, g, are established by using reliability theory, the factored loads are combined as discussed in Section 3.0 to create a maximum load effect.  A specific resistance factor, f, is then developed corresponding to the load combination(s) based on measured resistances and their computed variances from nominal resistances predicted by numerical models for resistance, e.g., the b-method for side friction of drilled shafts in sands.  Similar to the loads, the uncertainties in the resistances are quantified based on reliability (probability) theory.  The load and resistance factors include a consideration of the differences between measured and nominal values of the loads and resistances, respectively.

By using factored loads, γQ, and factored resistances, φR, the designer can establish a limit state, γ.     A limit state is a condition beyond which the bridge or component ceases to satisfy the provisions for which it was designed.  The limit state may be defined by linear (addition or subtraction) and/or non-linear (product or ratios) combinations of factored loads and factored resistances.  The linear version, γ = φR-γQ ≥ 0, is the most commonly used formulation of a limit state in the AASHTO-LRFD framework.  From practical considerations, an acceptable risk level is determined for each limit state, i.e., the probability that φR-γQ < 0, because otherwise the design for the case of φR-γQ ≥ 0 (i.e., no failure) will be very expensive.  Thus, in the AASHTO-LRFD approach, safety considerations are incorporated through load and resistance factors derived on the basis of an acceptable level of risk or acceptable probability of failure.  This process is in contrast to the traditional ASD approach (AASHTO, 2002) where safety is achieved with a single factor of safety applied to the resistance to obtain an allowable stress (or load).

It is important to realize that when the load and resistance factors are developed in the limit state concept as described above, they are completely tied to each other and form a pair.  In other words, neither the load nor the resistance factor can be changed unilaterally in the AASHTO-LRFD framework.  This does not mean that these factors cannot be changed based on local practices or past successful practices.  Rather it means that if one factor is changed, the owner/designer should perform the appropriate reliability-based calibration computations to determine the other factor.