# Improved Calculation of Load and Resistance Factors Based on Third-Moment Method
## 1 Introduction  
Load and resistance factor design (LRFD), as a rational probabilistic approach based on reliability to manage uncertainties in the building process, is widely employed in structural engineering [1; 2; 3; 4; 5; 6; 7]. In the past three decades, several methods for determining load and resistance factors have been developed, of which the first order reliability method (FORM) [8] was frequently used to account for stochastic fracture problems [9; 10; 11]. FORM is suggested to observe reliability accurately and generally. However, the design point must be firstly determined, and derivative-based iterations have to be used. Furthermore, the probability density functions (PDFs) of random variables are required. Thus, reliability calculated based on FORM is inefficient.  
In order to improve the computational efficiency and simultaneously guarantee accurate results, several simplified methods were proposed. According to the Mori method [12], all random variables are assumed to have known PDFs and are transferred into lognormal random variables. The results calculated by applying the Mori method were proved to be highly accurate. In the ASCE method [13], the load factors are well approximated by a simple function, which is valid for a broad range of common probability distributions used to model structural loads. Moment-based methods, being sufficiently accurate and efficient, have been widely used to assess the reliability analytically [14; 15; 16; 17; 18]. For example, the existing third-moment (3M) method [19] shows no need for the design point or any assumption of the PDFs of random variables. In this method, the iterative computation of the target mean resistance is simplified to one computation. Then, the load and resistance factors can be easily calculated.  
In this paper, the applications of the existing methods (including the ASCE method, the existing 3M method, and the Mori method) are inspected, and a simpler method is proposed. Compared with the existing 3M method [19], in which iterative-based computation is still inevitable, this paper provides a simplified method for the computation of target mean resistance without iteration, by improving the accuracy of computing the mean target resistance. Besides, the comparison of the ASCE method, the existing 3M method, and the proposed method is obtained by employing the load combination and example in the ASCE Standard 7-10 (2.3.2) [13]. The result shows that the ASCE method is easy and safe, but a waste of material.  
Generally, the difficulties with the existing methods can be summarized as: (1) for the existing 3M method, iterative-based computation is required. (2) There is a mathematical limitation for the existing 3M method, and its calculation formula is complicated. (3) For the ASCE method, the accuracy of calculation needs to be improved. (4) For the Mori method, the PDFs of some random variables are not always provided, resulting in calculation difficulties. Additionally, the random variables of the Mori method are required to be independent and obey the lognormal distribution.  
In light of the aforementioned analysis, it is necessary to develop a simple and accurate method. Therefore, in this paper, a new method for the calculation of load and resistance factors is proposed. The contribution of this paper is mainly reflected in the following points: (1) the calculation processes of three existing methods are summarized. (2) A new model is proposed to avoid the mathematical limitation and to overcome the shortcoming of iterative computation. (3) The feasibility of calculating target mean resistance without iteration of the proposed method is investigated. (4) Based on an example (ASCE 7-10, C2.3.6), the accuracy of the proposed method is investigated. (5) Nine cases with different distribution types are applied to verify the precision of the proposed method.