Estimation of load and resistance factors using the third-moment method based on the 3P-lognormal distribution

Abstract

Load and resistance factors are generally obtained using the first order reliability method (FORM) in which the design point should be determined and derivative-based iterations used. In this article, the third-moment reliability index, based on the three-parameter lognormal (3P-lognormal) distribution, is investigated. A simple method based on the third-moment method for estimating load and resistance factors is then proposed, and a simple formula for the target mean resistance is also presented to avoid iterative computations. Unlike the currently used method, the proposed method can be used to determine load and resistance factors, even when the probability density functions (PDFs) of the basic random variables are not available. Moreover, the proposed method does not require the iterative computation of derivatives or any design points. Thus, the method provides a more convenient and effective way to estimate load and resistance factors in practical engineering applications. Numerical examples are presented to demonstrate the advantages of the proposed third moment method for determining load and resistance factors.

Keywords

load and resistance factors / third-moment method / three-parameter lognormal (3P-lognormal) distribution / target mean resistance / simple formula

Introduction

Because a structure must perform under conditions of uncertainty, probabilistic analyses are generally necessary for reliability-based structural designs. However, a reliability-based structural design may also be achieved without a complete probabilistic analysis. If the required safety factors are predetermined on the basis of specified probability-based requirements, reliability-based designs can be accomplished through the adoption of appropriate deterministic design criteria, such as the use of traditional safety factors.

For practical use, design criteria should be as simple as possible; moreover, they should be developed in a way that is familiar to practicing engineers. This can be accomplished through the use of load amplification factors and resistance reduction factors, known as the LRFD format [1,2]. The nominal design loads are amplified by appropriate load factors, and the nominal resistances are reduced by the corresponding resistance factors. Safety is assured if the factored resistance is at least equal to the factored loads. The appropriate load and resistance factors must be developed to obtain designs that achieve a prescribed level of reliability.

Load and resistance factors are generally determined using the first order reliability method (FORM) (e.g., Ref. [3]) in which the “design point” must be determined and derivative-based iterations have to be used. In general, practicing engineers only use the load and resistance factors recommended in design codes. However, with the trend toward performance design, there will be a necessity for designers to determine the load and resistance factors by themselves in order to conduct structural design more flexibly design structures. In such a case, the design code is required to recommend not only specific load and resistance factor values but also suitable and simple methods for determining these factors. AIJ [2] provided a simple method based on the work of Mori [4] in which all random variables are assumed to have known probability density functions (PDFs) and are required to transfer into lognormal random variables. However, in reality, the PDFs of some basic random variables are often unknown due to a lack of statistical data. Therefore, obtaining an LRFD that includes random variables with unknown PDFs is important.

In this article, the basic principle of the load and resistance factor format is reviewed. The third-moment reliability index, based on the three-parameter lognormal (3P-lognormal) distribution, is investigated. A simple method based on the third-moment method for estimating the load and resistance factors is proposed, and a simple formula for the target mean resistance is also presented to avoid iterative computations. Because the proposed method is based on the first three moments of the basic random variables, the load and resistances factors can be determined even when the probability density functions of the random variables are unknown. The simplicity and efficiency of the proposed method for determining load and resistance factors are demonstrated with several numerical examples.

Review of load and resistance factor determination

The LRFD format can be expressed as follows

(1)\varphi R_{\mathrm{n}}\ge \mathrm{\sum }{\gamma }_i{S}_{\mathrm{n}i},

where \varphi is the resistance factor; γ_i is the partial load factor to be applied to the load S_i; R_n is the nominal resistance value; and S_ni is the nominal value of load S_i.

In reliability-based structural designs, the resistance factor \varphi and the load factor γ_i should be determined based on a specified reliability; that is, the design format, Eq. (1), should be probabilistically equivalent to the following equations.

(2)G(X)=R-\mathrm{\sum }{S}_i,

(3)P_{\mathrm{f}}\le P_{\mathrm{f}\mathrm{T}}\mathrm{o}\mathrm{r}\beta \ge {\beta }_{\mathrm{T}},

where R and S_i are random variables representing uncertainty in the resistance and load effects, respectively; P_f and β are the probability of failure and the reliability index, respectively, corresponding to the performance function Eq. (2); and P_fT and β_T are the target probability of failure and the target reliability index, respectively.

If R and S_i are mutually independent normal random variables, the second moment method is correct and the design formula becomes

(4){\beta }_{\mathrm{2}M}\ge {\beta }_{\mathrm{T}},

where

(5a){\beta }_{\mathrm{2}M}={\displaystyle \frac{{\mu }_G}{{\sigma }_G}},

(5b){\mu }_G={\mu }_R-\mathrm{\sum }{\mu }_{S_i},{\sigma }_G=\sqrt{{\sigma }_R^{\mathrm{2}}+\mathrm{\sum }{\sigma }_{S_i}^{\mathrm{2}}}.

Here, β_2M is the second moment reliability index; μ_G and σ_G are the mean value and standard deviation of the performance function G(X), respectively; μ_R and σ_R are the mean value and standard deviation of R, respectively; and μ_Si and σ_Si are the mean value and standard deviation of S_i, respectively.

Substituting Eq. (5) in Eq. (4) leads to

(6){\mu }_R(\mathrm{1}-{\alpha }_R{V}_R{\beta }_{\mathrm{T}})\ge \mathrm{\sum }{\mu }_{S_i}(\mathrm{1}+{\alpha }_{S_i}{V}_{S_i}{\beta }_{\mathrm{T}}),

Comparing Eq. (6) with Eq. (1), the load and resistance factors can be expressed as

(7a)\varphi =(\mathrm{1}-{\alpha }_R{V}_R{\beta }_{\mathrm{T}}){\displaystyle \frac{{\mu }_R}{R_{\mathrm{n}}}},

(7b){\gamma }_i=(\mathrm{1}+{\alpha }_{S_i}{V}_{S_i}{\beta }_{\mathrm{T}}){\displaystyle \frac{{\mu }_{S_i}}{S_{\mathrm{n}i}}},

where V_R and V_Si are the coefficients of variation for R and S_i, respectively; and α_R and α_Si are the direction cosines, or separating factors, of R and S_i, respectively.

(8){\alpha }_R={\displaystyle \frac{{\sigma }_R}{{\sigma }_G}},{\alpha }_{S_i}={\displaystyle \frac{{\sigma }_{S_i}}{{\sigma }_G}}\mathrm{.}

When R and S_i are non-normal random variables, the second moment reliability index expressed in Eq. (5) does not correctly reflect the real failure probability corresponding to the performance function in Eq. (2). In this case, the reliability index is generally obtained using the FORM, and the design format can be expressed as

(9)R^{*}\ge \mathrm{\Sigma }{S}_i^{*}\mathrm{.}

The load and resistance factors can be obtained as [1]

(10)\varphi ={\displaystyle \frac{R^{*}}{R_{\mathrm{n}}}},{\gamma }_i={\displaystyle \frac{S_i^{*}}{S_{\mathrm{n}i}}},

where R* and S_i^{\ast } are the values of the variables R and S_i, respectively, at the design point of the FORM. Because R* and S_i^{\ast } are usually obtained using derivative-based iterations, explicit expressions of R* and S_i^{\ast } are not available. Some simplifications have been proposed to avoid iterative computations [4-6].

In this article, the reliability index based on the 3P-lognormal distribution is used as the reliability index for the performance function of Eq. (2). Because the central moments of the performance function, as described in Eq. (2), can be obtained easily, the probability of failure, defined as P[G(X)<0], can be expressed as a function of the central moments [7], and because no derivative-based iterations are necessary in the proposed method, the required load and resistance factors are much easier to determine.

Third-moment reliability index based on the 3P-lognormal distribution

3P-lognormal distribution

For the standardized variable x_s = (x-μ)/σ, the probability density function of the 3P-lognormal distribution has been defined as [8]

(11)f(x_{\mathrm{s}})={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathrm{\pi }\ln (A)}\left|x_{\mathrm{s}}-u_{\mathrm{b}}\right|}}\exp \left\{-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\ln (A)}}{\left[\ln \left(\sqrt{A}{\displaystyle \frac{\left|x_{\mathrm{s}}-u_{\mathrm{b}}\right|}{\left|u_{\mathrm{b}}\right|}}\right)\right]}^{\mathrm{2}}\right\},

where

(12)A=\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{u_{\mathrm{b}}^{\mathrm{2}}}},

here, μ and σ are the mean and standard deviation of x, respectively, and u_b is the standardized bound of the distribution.

The relationship between x_s and u is given by

(13)x_{\mathrm{s}}=u_{\mathrm{b}}\left\{\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\sqrt{A}}}\exp [\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}({\alpha }_{\mathrm{3}})\sqrt{\ln (A)}u]\right\},

where α₃ is the third dimensionless central moment, or the skewness of x; and Sign(α₃) is -1, 0 or 1 if α₃ is negative, zero or positive, respectively. The relationship between the bound u_b and the skewness α₃ is given by

(14){\alpha }_{\mathrm{3}}=-\left(\mathrm{3}+{\displaystyle \frac{\mathrm{1}}{u_{\mathrm{b}}^{\mathrm{2}}}}\right){\displaystyle \frac{\mathrm{1}}{u_{\mathrm{b}}}}\mathrm{.}

The solution of Eq. (14) yields

(15)u_{\mathrm{b}}={(a+b)}^{\frac{\mathrm{1}}{\mathrm{3}}}+{(a-b)}^{\frac{\mathrm{1}}{\mathrm{3}}}-{\displaystyle \frac{\mathrm{1}}{{\alpha }_{\mathrm{3}}}},

and

(16)a=-{\displaystyle \frac{\mathrm{1}}{{\alpha }_{\mathrm{3}}}}\left({\displaystyle \frac{\mathrm{1}}{{\alpha }_{\mathrm{3}}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\right),b={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\alpha }_{\mathrm{3}}^{\mathrm{2}}}}(\sqrt{{\alpha }_{\mathrm{3}}^{\mathrm{2}}+\mathrm{4}})\mathrm{.}

The PDFs of the standard 3P-lognormal distribution for α₃<0 and α₃>0 are shown in Fig. 1. When α₃ = 0, the PDF degenerates to that of the standard normal distribution and is depicted as a thick solid line in Fig. 1. As shown in this figure, the distribution reflects the characteristics of skewness.

The simplification of the distribution

According to Eq. (13), the absolute value of the random variable |x_s–u_b| obeys the lognormal distribution with parameters λ and ζ, and the coefficient of skewness α₃ is given by

(17){\alpha }_{\mathrm{3}}=[\exp ({\zeta }^{\mathrm{2}})+\mathrm{2}]\sqrt{\exp ({\zeta }^{\mathrm{2}})-\mathrm{1}}\mathrm{.}

Comparing Eq. (17) with Eq. (14) yields

(18)u_{\mathrm{b}}={\displaystyle \frac{\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}({\alpha }_{\mathrm{3}})}{\sqrt{\exp ({\zeta }^{\mathrm{2}})-\mathrm{1}}}},

when |α₃|<1, the absolute value of ζ is less than 0.314, and the following approximation applies, with an error of less than 2.5%:

(19)\sqrt{\exp ({\zeta }^{\mathrm{2}})-\mathrm{1}}=\zeta .

Substituting Eq. (19) into Eq. (17) yields

(20){\alpha }_{\mathrm{3}}=\mathrm{3}\zeta +{\zeta }^{\mathrm{3}}.

For small ζ values, such as ζ<0.314, Eq. (20) yields the following approximations for ζ and u_b:

(21)u_{\mathrm{b}}=-{\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}}}},\zeta ={\displaystyle \frac{{\alpha }_{\mathrm{3}}}{\mathrm{3}}}.

To examine the accuracy of the approximation expressed in Eq. (21), the values of u_b are depicted in Fig. 2, in which the thin solid lines indicate the exact values obtained from Eq. (15) and the thick dashed lines indicate those obtained with Eq. (21). As shown in the figure, the u_b results obtained with Eq. (21) agree well with those obtained from Eq. (15).

The third-moment reliability index

For a performance function z = G(X), without loss of generality, G(X) can be standardized as follows:

(22)z_u={\displaystyle \frac{(z-{\mu }_G)}{{\sigma }_G}},

where μ_G and σ_G are the mean value and standard deviation of G(X), respectively. For a performance function z = G(X), if the first three moments are obtained, the standardized variable z_u defined by Eq. (22) is assumed to obey the simplified 3P-lognormal distribution, and the standard normal random variable u can be expressed as the following function:

(23)u={\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{6}}}+{\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}G}}}\ln \left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\alpha }_{\mathrm{3}G}{z}_u\right),

because

(24)P_{\mathrm{f}}=P[z=G(X)\le \mathrm{0}]=P[z_u\le -{\beta }_{\mathrm{2}M}],

the third-moment reliability index based on Eq. (23) is obtained as

(25){\beta }_{\mathrm{3}M}=-{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{6}}}-{\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}G}}}\ln \left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\alpha }_{\mathrm{3}G}{\beta }_{\mathrm{2}M}\right),

(26)P_{\mathrm{f}}=-{\mathrm{\Phi }}^{-\mathrm{1}}(-{\beta }_{\mathrm{3}M}),

where β_3M is the third-moment reliability index.

Note that when α_3G = 0, Eq. (25) cannot directly generate appropriate results. Because the limit value of Eq. (25) is 0 for α_3G≥0, the third-moment reliability index is expressed directly as

(27){\beta }_{\mathrm{3}M}={\beta }_{\mathrm{2}M}\mathrm{f}\mathrm{o}\mathrm{r}{\alpha }_{\mathrm{3}G}=\mathrm{0.}

Load and resistance factors using the third-moment method

Substituting the third-moment reliability index in the design format described in Eq. (3) gives

(28){\beta }_{\mathrm{3}M}=-{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{6}}}-{\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}G}}}\ln \left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\alpha }_{\mathrm{3}G}{\beta }_{\mathrm{2}M}\right)\ge {\beta }_{\mathrm{T}},

The α_3G in Eq. (2) can be computed as

(29){\alpha }_{\mathrm{3}G}={\displaystyle \frac{\mathrm{1}}{{\sigma }_G^{\mathrm{3}}}}({\alpha }_{\mathrm{3}R}{\sigma }_R^{\mathrm{3}}-{\displaystyle \sum {\alpha }_{\mathrm{3}{S}_i}{\sigma }_{S_i}^{\mathrm{3}}}),

where α_3R and α_3Si are the sknewness values of R and S_i, respectively.

From Eq. (28),

(30){\beta }_{\mathrm{2}M}\ge {\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}G}}}\left\{\mathrm{1}-\exp \left[{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{3}}}\left(-{\beta }_T-{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{6}}}\right)\right]\right\}.

Denoting the right side of Eq. (30) as β_2T,

(31){\beta }_{\mathrm{2}M}\ge {\beta }_{\mathrm{2}\mathrm{T}},

(32){\beta }_{\mathrm{2}\mathrm{T}}={\displaystyle \frac{\mathrm{3}}{{\alpha }_{\mathrm{3}G}}}\left\{\mathrm{1}-\exp \left[{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{3}}}\left(-{\beta }_{\mathrm{T}}-{\displaystyle \frac{{\alpha }_{\mathrm{3}G}}{\mathrm{6}}}\right)\right]\right\}.

Equation (31) is a strictly derived design formula equivalent to Eq. (3), which implies that if the second moment reliability index β_2M is at least equal to β_2T, the reliability index β will be at least equal to the target reliability index β_T, and the required reliability is satisfied. Therefore, β_2T can be considered to be a target value of β_2M and is hereafter denoted as the target second moment reliability index.

Substituting Eq. (5) into Eq. (31) gives

(33){\mu }_R(\mathrm{1}-{\alpha }_R{V}_R{\beta }_{\mathrm{2}\mathrm{T}})\ge \mathrm{\sum }{\mu }_{S_i}(\mathrm{1}+{\alpha }_{S_i}{V}_{S_i}{\beta }_{\mathrm{2}\mathrm{T}}).

Comparing Eq. (33) with Eq. (1), the load and resistance factors can be expressed as

(34a)\varphi =(\mathrm{1}-{\alpha }_R{V}_R{\beta }_{\mathrm{2}\mathrm{T}}){\displaystyle \frac{{\mu }_R}{R_{\mathrm{n}}}},

(34b){\gamma }_i=(\mathrm{1}+{\alpha }_{S_i}{V}_{S_i}{\beta }_{\mathrm{2}\mathrm{T}}){\displaystyle \frac{{\mu }_{S_i}}{S_{\mathrm{n}i}}},

where V_R and V_Si are the coefficients of variation, and α_R and α_Si are the direction cosines for R and S_i, respectively, which are the same as those in Eq. (7).

Comparing Eq. (34) with Eq. (7), after replacing β_T in Eq. (7) by β_2T in Eq. (34), the method for determining the load and resistance factors using the third moment method is essentially the same as the second moment method.

The variations of the target second moment reliability index β_2T with respect to the target reliability index β_T are shown in Fig. 3. In this figure, β_2T is larger than β_T for negative α_3G and smaller than β_T for positive α_3G.

Determination of the mean resistance

Because the load and resistance factors are determined when the reliability index is equal to the target reliability index, the mean resistance value should be determined under this condition, which is hereafter referred to as the target mean resistance. Generally, the target mean resistance is computed using the following iteration equation [9]:

(35){\mu }_{Rk}={\mu }_{Rk-\mathrm{1}}+({\beta }_{\mathrm{T}}-{\beta }_{k-\mathrm{1}}){\sigma }_G,

where μ_Rk and μ_Rk-1 are the kth and (k-1)th iteration value of the mean resistance value, respectively, and β_k-1 is the (k-1)th iteration value of the third-moment reliability index.

To avoid iterative computations of the target mean resistance, the following simple formula is proposed:

(36){\mu }_{R\mathrm{T}}=\mathrm{\Sigma }{\mu }_{S_i}+{\beta }_{\mathrm{2}{\mathrm{T}}_{\mathrm{0}}}{\sigma }_{G_{\mathrm{0}}},

where μ_RT is the target mean resistance; {\sigma }_{G_{\mathrm{0}}} is the standard deviation of G(X); and β_2T0 is the target 2M reliability index, which is obtained using μ_R0.

μ_R0 is given by the following equation, which was obtained from trial and error:

(37){\mu }_{R_{\mathrm{0}}}=\mathrm{\sum }{\mu }_{S_i}+\sqrt{{\beta }_{\mathrm{T}}^{\mathrm{3.5}}\mathrm{\sum }{\sigma }_{S_i}^{\mathrm{2}}}.

The steps for determining the load and resistance factors using the third moment method are as follows.

1) Calculate μ_R0 using Eq. (37).

2) Calculate σ_G0, α_3G0 and β_2T0 using Eq. (5), Eq. (29) and Eq. (32), respectively, and determine μ_RT with Eq. (36).

3) Calculate σ_G, α_3G and β_2T using Eq. (5), Eq. (29) and Eq. (32), respectively, and calculate α_R and α_Si with Eq. (8).

4) Determine the load and resistance factors using Eq. (34).

Numerical examples

Example 1

Consider the following performance function:

(38)G(X)=R-(D+L),

where R = resistance, a lognormal variable with μ_R/R_n = 1.1 and V_R = 0.15; D = dead load, a normal variable with μ_D/D_n = 1.0 and V_D = 0.1; and L = live load, a lognormal variable with μ_L/L_n = 0.45 and V_L = 0.4.

Consider the mean value of D and L with μ_D = 1.0 and μ_L/μ_D varying from 0.5 to 4.0. Determine the target mean resistance and the corresponding load and resistance factors for the given target reliability index β_T = 2.5.

The load and resistance factors obtained using the present third-moment method are illustrated in Fig. 4(a) for β_T = 2.5, and they are compared with the corresponding factors obtained with the FORM and the second-moment method. The target mean resistance values obtained using the proposed method and those obtained using the FORM and the second-moment method are illustrated in Fig. 4(b). From Fig. 4, although the load and resistance factors obtained using the proposed method are different from those obtained using the FORM, the target mean resistance values obtained using the proposed method are essentially the same as those obtained using the FORM. However, for the second-moment method, when μ_L/μ_D is small, the proposed method can provide good results. When μ_L/μ_D becomes larger, the method will yield unconservative results.

Example 2

Consider the following performance function:

(39)G(X)=R-(D+L+W),

where R = resistance, a lognormal variable with μ_R/R_n = 1.1 and V = 0.15; D = dead load, a normal variable with, μ_D/D_n = 1 and V = 0.1; L = live load, a lognormal variable with μ_L/L_n = 0.45 and V = 0.4; and W = wind load, a Gumbel variable with μ_W /W_n = 0.6 and V = 0.2.

Consider the mean values of D, L, and W with μ_D = 1.0, μ_L/μ_D = 0.5, and μ_W/μ_D = 2.0. Determine the target mean resistance and the corresponding load and resistances factors for any given target reliability index β_T.

The load and resistance factors obtained using the proposed third-moment method are illustrated in Fig. 5(a), and they are compared with the corresponding factors obtained using the FORM and the second-moment method. The target mean resistances obtained using the proposed third-moment method and those obtained using the second-moment method and the FORM are illustrated in Fig. 5(b). As shown in this figure, although the load and resistance factors obtained using the proposed third-moment method are different from those obtained using the FORM, the target mean resistance values obtained using the proposed method are essentially the same as those obtained using the FORM. However, for the second-moment method, when β_T is small, the method can provide good results. When β_T becomes larger, it will yield incorrect results.

Example 3

Consider the following performance function:

(40)G(X)=R-(D+L+S+W)

where R = resistance, a lognormal variable with μ_R/R_n = 1.1 and V = 0.15; D = dead load, a normal variable with, μ_D/D_n = 1 and V = 0.1; L = live load, a lognormal variable with μ_L/L_n = 0.45 and V = 0.4; S = snow load, a Gumbel variable with μ_S /S_n = 0.47 and V = 0.25; and W = wind load, a Gumbel variable with μ_W /W_n = 0.6 and V = 0.2.

Consider the mean values of D, L, S and W with μ_D = 1.0, μ_L/μ_D = 0.5, μ_S/μ_D = 2.0 and μ_W/μ_D = 2.0. Determine the target mean resistance values for any given target reliability index β_T.

The load and resistance factors obtained using the proposed third-moment method are illustrated in Fig. 6(a), and they are compared with the corresponding factors obtained using the FORM and the second-moment method. The target mean resistance values obtained using the proposed third-moment method and those obtained using the second-moment method and the FORM are illustrated in Fig. 6(b). As shown in this figure, although the load and resistance factors obtained using the present third-moment method are different from those obtained using the FORM, the target mean resistance values obtained using the proposed method are essentially the same as those obtained using the FORM. Therefore, in design practice, if the resistance factors determined from either of the methods are adopted, the corresponding load factors (i.e., the factors determined from the same method as the one used to estimate the resistance factors) should be used. Similar to example 1 and example 2, when β_T is small, the second-moment method can provide good results. When β_T becomes larger, it will yield incorrect results.

Conclusions

1) The third-moment reliability index based on the three-parameter lognormal distribution was derived. A method for determining load and resistance factors using the third-moment method was proposed, and a simple formula for the target mean resistance was also presented to avoid iteration computations. Derivative-based iterations, which are necessary in the FORM, were shown to be unnecessary in the proposed method. Therefore, the proposed method is much easier to apply.

2) Although the load and resistance factors obtained with the proposed method may be different from those obtained with the FORM, the target mean resistances obtained with both methods are essentially the same.

3) Because the proposed method is based on the first three moments of the basic random variables, load and resistances factors can be determined even when the probability density functions of the basic random variables are unknown.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant No. 51008313), the Sheng-hua Lie-ying Program of Central South University, a grant from the National High Technology Research and Development Program of China (863 Program, No. 2009AA11Z101), and the Joint Research Fund for Overseas Chinese, Hong Kong and Macao Young Scholars (No. 50828801) from the National Natural Science Foundation of China. These supports are gratefully acknowledged.

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