Improving Confidence Intervals for the Resistance Factor in Load and Resistance Factor Design (LRFD)

Uncertainties inherent in the designs of construction are generally quantified as loads and resistances. Loads are forces applied to structures which cause stress. Dead Loads Q_DL and Live Loads Q_LL are the two main types of loads considered. The resistance R is the capacity of the structure to bear the loads. For safe use of the structure, resistance must exceed the loads. Load and Resistance Factor Design (LRFD) is a reliability-based design used by structural engineers which considers the loads and the resistance to be independent random variables. LRFD aims to design structural elements with a target probability of failure by trying to prevent limit states via load factors γ_DL, γ_LL and resistance factor ϕ. A reliability index β is typically used to express reliability. The lognormal distribution is often assumed for both loads and resistances. The resistance factor ϕ has to be estimated to achieve a specified probability of failure, which is called resistance factor calibration. In LRFD under the assumption of log normal distribution, when dead load, live load, and resistance are considered, a simple closed form solution cannot be found for the resistance factor, so the resistance factor is usually estimated to achieve a target probability of failure via First Order Second Moment (FOSM) approximation or Monte Carlo Simulation (MCS). The estimate of the resistance factor ϕ has been treated as a fixed constant. But since limited samples are used to estimate the resistance factor, it is only a point estimate with sampling error. If the variability is not considered, the desired probability of failure might not be achieved therefore it is more appropriate to report confidence intervals for the resistance factor. Confidence intervals for the resistance factor using FOSM which are provided in the literature demonstrate significantly low coverage compared to the nominal level. In this research confidence intervals are constructed using MCS. A method of tuning the confidence intervals for the resistance factor is explored to attain near nominal coverages. Simulations are conducted to identify the tuning factor needed to achieve 95% nominal coverage for parametric bias corrected bootstrap confidence intervals for the resistance factor using both estimation methods FOSM and MCS. Models are derived to estimate the tuning factor that should be used for given values of sample size n, coefficient of variation of resistance COV_R. These models are evaluated using published examples which reflect real life situations. Literature does not provide a clear explanation as to how FOSM approximation was derived. In order to provide clarity to the process, an alternative derivation of FOSM is presented in a step-by-step approach explicitly laying out the assumptions which are implicit in the process. It is discovered that the load and resistances factors play little to no role in the application of LRFD which is a consequence using the FOSM approximation. The relationship between the factor of safety FS, and the probability of failure of a structural element is explored which enables quantification of uncertainty with regard to Allowable Stress Design (ASD).