Method for Inferring the Design Value of the Resistance Based on Probabilistic Model

Methods for inferring the design value of the resistance based on test have long been studied extensively, but the existing methods have several limitations on unified guarantee rate ensurance and reliability control. Firstly, the rationales and deficiencies of the present methods in ISO 2394 : 2015 and EN 1990 : 2002 were generalized. Secondly, in view of the disadvantages, a new inferring method combining the probability model of resistance with statistical approach was put forward. *e proposed method established a relationship among design resistance, probability characteristics of known factors, and statistical results of unknown factors and possessed a rigorous and sound theoretical basis on both conditions that the coefficient of variation of model uncertainty was unknown and full known. Lastly, a contrast work was carried out between the Eurocodemethod and the proposed method; the results showed that the latter method had a higher inferring value, which means a better inferring result.


Introduction
e method for structure design based on test, aiming at ultimate limit state, was proposed in ISO 2394 : 1998 [1] and EN 1990 : 2002 [2] consecutively. Owing to its practicability and applicability to large batch of similar components, major or new structures, this method is adopted in GB 50153-2008 [3] and ISO 2394 : 2015 [4] and has become one of the major trends of structural design. e experiment-aided design method combines experimental data and statistical approach to infer the design value of resistance (i.e., design resistance). e inference is usually based on resistance variable, which relates to the test results only, or based on resistance analysis model, which adopts probability characteristics of influence factors. e latter method reduces the uncertainty to a large extent and obtains a better inference result [1,4], thus it has higher design value. For instance, the ISO 2394 : 2015 method has been introduced in Refs. [5][6][7] and EN 1990 : 2002 in Refs. [8,9], while the detailed deducing processes are different. ey are approximate and empirical separately, having certain limitations on the control of structural reliability.
Due to the limitations of the abovementioned two methods, an inferring method on the basis of the probability model of resistance and statistical theory is put forward which provides the ensurance of structural reliability control.

Method in ISO 2394 : 2015.
e design resistance of a structural element obtained from test, whose direct basis is the analysis model, can be calculated as follows [4]: of differences between reality and test conditions and can be determined by experience directly; θ d is the design value of the model uncertainty θ and should be deduced by test results.
In this international standard, θ is assumed as a lognormal distribution and its design value θ d can be inferred from the Bayesian estimation method based on Jeffreys noninformative prior distribution [4,10,11]: where x i and R i are the experimental values of X and R, respectively, which have to be measured in every experiment; w i is the value of W used in test; θ i is the value of θ decided by test, i � 1, 2, . . . , n, and n is the number of specimen; t vd is the Φ(β R ) quantile of the student distribution with a freedom degree of n − 1, where Φ(·) is the standard normal distribution function, β R � α d β, where β is the target reliability index and α d is the design value of sensitivity factor according to first-order reliability method (FORM). If the uncertainty of resistance is dominating during reliability analysis, α d = 0.8; if not, α d = 0.3. e abovementioned method can be used in both the design value method and partial factor method. In the former, the guaranteed rate of the design resistance is Φ(β R ) [4], while in the latter, this rate should be determined by probability analysis with β R replaced by Φ −1 (p d ).
To ensure the accuracy of reliability control, design resistance should adopt a uniform guarantee rate (i.e., Φ(β R ) or p d ) in the abovementioned two methods. While the implied rates are not all Φ(β R ) or p d in fact for the reason that the model reflecting the relationship between resistance and influence variables is only approximate. Of particular note is that Φ(β R ) is obtained just from the design value of model uncertainty in the partial factor method, resulting in a lower rate since many other influence variables exist, causing a higher referring value.

Method in EN 1990 : 2002.
A probability model is used to calculate the design resistance in EN 1990 : 2002; furthermore, the weighted average method of the design resistance under two ideal conditions (i.e., no prior knowledge and full knowledge of the coefficient of variation) is applied.
Assuming that the resistance follows a lognormal distribution LN(μ ln R , σ 2 ln R ), the design resistance R d can be expressed as the following equation: where μ R and V R are the mean and coefficient of variation of the resistance; p is the guarantee rate of R d . Supposing that R 1 , R 2 , . . . , R n are the samples of resistance, the inference of R d can be expressed as equation (7) when the coefficient of variation V R (which means the distribution parameter σ ln R ) is unknown and equation (8) when V R is known according to the Bayesian estimation method [2]: where m ln R and s ln R are the realized values of the mean and coefficient of variation of samples ln where n p is the p quantile of the standardized normal distribution. e design resistance using the probability model can be expressed as the following equation [2]: where g rt (·) is the theoretical model of resistance; b is the ratio between the experimental value and the theoretical value obtained by least squares regression; the mean value of model uncertainty θ is about 1.
Assuming that θ and g rt (X) obey lognormal distributions LN(μ ln θ , σ 2 ln θ ) and LN(μ ln rt , σ 2 ln rt ) separately, the following simplification can be made: with where V θ and V rt are the coefficient of variation of θ and g rt (X) separately. In most cases, Q is known partly for Q θ being unknown, while Q rt is full known. erefore, the inferring value of design resistance ranges from the value obtained by no prior knowledge about Q to full knowledge of Q. Moreover, with the rising of the ratio of Q 2 θ to Q 2 , the inferring result will get closer to the case where Q is unknown, from the value obtained by no prior knowledge about Q to full knowledge of Q. us, the weighting factors can take the following expressions: After weighted average of the inferring results was obtained by equations (7) and (8), the following equation can be obtained: Assuming the following approximate equations are available, the inference design resistance can be expressed as follows [2]: where m R is the realized value of mean of samples R 1 , R 2 , . . . , R n ; μ X is the vector of mean of influence factors of resistance; m θ is the realized value of mean of samples θ 1 , θ 2 , . . . , θ n , when fitting the test results by the least squares method; the value is 1.
If the coefficient of variation of θ is known, the inference design resistance can be expressed as the following equation [2]: It is experienced to adopt the weighted average method to infer the design resistance in EN 1990 : 2002. More seriously, there are four drawbacks in the abovementioned method: Firstly, k ∞ should be n p ������ 1 + 1/n √ , not n p in equations (18) and (19), for large errors will be produced when the number of specimen is not large; Secondly, the relationship between m ln R and m R in equation (17) is not valid when the number of experiments is small, while the available relationship should be expressed as follows: where μ ln R and μ R are the true values of m ln R and m R , respectively; irdly, equations (7) and (8) respectively represent the conditions that the coefficient of variation of resistance are unknown and full known, where s ln R is a statistical value and σ ln R is a given distribution parameter. However, in the process of establishing equation (18), distribution parameter Q is determined by s ln θ in accordance with equation (15), which does not represent a full known condition.
Finally, equation (16) which calculates the statistical value s ln R does not conform to the calculation of standard deviation in statistics. e correct equation should be expressed as follows [12]: where s ln θ, ln rt is the covariance of ln θand ln g rt (X); s ln rt is the standard deviation of ln g rt (X). ese four drawbacks are necessary approximations to calculate the design resistance, but not rigorous in theory.

An Inferring Method Based on the Probability
Model of Resistance e inferring method based on the probability model builds the relationship among design resistance, probability characteristics of known factors, and statistical results of unknown factors by statistical theory. is kind of model has also been used in EN 1990 : 2002, while just in analyzing the probability characteristics of resistance. us, the inferring method of Eurocode is based on the resistance variable in essence.
e logarithm of design resistance can be expressed as follows in the probability model inferring method: e approximate equation is available within a practical scope for 0.05 ≤ V θ ≤ 0.25 and 0.05 ≤ V rt ≤ 0.25: Equation (23) can be regarded as an accurate formula with a range of absolute errors from −0.005 to 0.008 and relative errors from −0.019 to 0.029 for the errors are small. By assigning this to equation (22), we obtain lnR d � min μ lnθ − 0.9kQ θ + lnb + lng rt X m − 0.5kQ rt , Taking Mathematical Problems in Engineering where ln θ and S ln θ are the mean and coefficient of variation of samples ln θ 1 , ln θ 2 , . . . , ln θ n separately, T 1 and T 2 follow the noncenter student distribution with a freedom degree of both n − 1 and noncentrality parameters of 0.9k � n √ and 0.5k � n √ , respectively [13]. e design resistance can be expressed as equation (26) when (s ln θ /Q rt ) ≥ (0.4k/k n,0.9 − k n,0.5 ) or equation (26) when (s ln θ /Q rt ) < (0.4k/k n,0.9 k n,0.5 )by the method of interval estimation. R d � m θ ′ bg rt μ X exp − k n,0.9 s lnθ + 0.5kQ rt , m θ ′ bg rt μ X · exp − k n,0.5 s ln θ + 0.9kQ rt .
(26) e realized value of the mean of samples θ 1 , θ 2 , . . . , θ n , k n,0.9 , and k n,0.5 can be calculated in the following expressions: Mathematical Problems in Engineering where t (n−1,0.9k � n √ ,C) and t (n−1,0.5k � n √ ,C) are the C quantiles of the noncenter student distribution with a freedom degree of both n − 1 and noncentrality parameters of 0.9k � n √ and 0.5k � n √ , respectively, whose values can be supplied in the form of a table. C is the confidence level with a desirable value for 0.75 [4,[14][15][16][17].
T ′ , subjected to the standardized normal distribution, can be taken as the following expression when the coefficient of variation of θ is known: e design resistance can be expressed as equation (29) by the method of interval estimation: where n C is the C quantile of the standardized normal distribution. In normal case, the coefficient of variation of θ is unknown. If a larger value of V θ (or Q θ ) is adopted by experience, a better inferring result will be acquired from equation (29) than equation (26), which means a higher referring design value is obtained. e abovementioned method, which is based on the probability model of resistance and builds a relationship among design resistance, probability characteristics of known factors, and statistical results of unknown factors by statistical theory directly, has a rigorous and sound theoretical basis for both conditions where the coefficient of variation of θ is unknown and full known.  Figure 1 shows the differences between the EN 1990 : 2002 method and the proposed method when the coefficient of variation of θ is known, and Figure 2 shows the differences when the coefficient of variation of θ is unknown. In order to facilitate the comparison, set

Comparison and Analysis
From Figures 1 and 2, it can be seen that (1) e inferring results are better when the coefficient of variation of θ is known rather than unknown, especially when the number of specimen is small. ideal conditions (i.e., no prior knowledge and full knowledge of the coefficient of variation), which is a simplified and experienced method merely, while the proposed method is established on the basis of statistics strictly, thus it has a more accurate outcome.

Conclusion
is study provides a new inferring method of design resistance based on the probability model of resistance, and the proposed method is superior to the method in ISO 2394 : 2015 and EN 1990 : 2002. Based on the analysis and comparison, several clear conclusions can be drawn as follows: (1) e inferring method presented in ISO 2394 : 2015 which is based on the analysis model reflects an approximate relationship between resistance and the influence variables from the viewpoint of probability. By this method, a unified guarantee rate cannot be ensured, which will affect the reliability control to some extent. It is also worth noting that the method takes the guarantee rate of model uncertainty as the implied rate of design resistance, causing it to be lower than the specified one. (2) e inferring method put forward in EN 1990in EN : 2002 which is based on the probability model in the process of analyzing the probability characteristic of resistance, is a method on the basis of resistance variable. In addition, it is empirical to use the weighted average method and approximate calculations. (3) e proposed method, which is based on the probability model of resistance, establishes a relationship among design resistance, probability characteristics of known factors, and statistical results of unknown factors by statistical theory directly and has a rigorous and sound theoretical basis. (4) e case comparison shows that the discrepancy between the values acquired from EN 1990 : 2002 and the proposed method is small when the coefficient of variation of θ is known, while it becomes wider when the coefficient of variation of θ is unknown. Furthermore, when the specimen number is small, the inferring results are lower, even reducing to 0 by the EN 1990 : 2002 method; while it is higher by the proposed method, it means better inferences.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 6 Mathematical Problems in Engineering