# Comment and Improvement on the Methods for Deriving Structural Resistance Design Value

Authors: Hu Yukun; Yao Jitao; Ma Pengfei; Cheng Zhengjie; Zhang Luyang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

With the increasing emphasis on sustainability, more and more environmentally friendly new materials and structures are developed, but they are difficult to be applied in engineering practice due to the lack of mature force resistance models. The design assisted by the testing method proposed by ISO2394:2015 and EN1990:2002 can solve this problem effectively. The aim of this paper was to analyze the characteristics of the method from a mathematical and statistical point of view based on the data and to suggest improvements. The resistance of each concrete member was derived by the design assisted by testing methods. The derivation results showed that thep values, the number of tests n, the coefficient of variation θ, and the coefficient of variation of resistance VR known or unknown have a large impact on the derived results. Also, the derivation by the Bayesian procedure or interval estimation methods might be negative. This indicates that the theory of the method is not rigorous and has some disadvantages.


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## Body

## 1. Introduction

Sustainable development has attracted worldwide attention during the past two decades [1, 2]. As a major contributor to the world’s energy consumption and carbon emissions, sustainable buildings and construction industry have been emphasized [3–6]. While operational emissions owing to the considerable energy are generally assumed to play a key role in reducing building embodied carbon emissions [7–9], more recent attention has been given to sustainable buildings’ components and materials [10–12].However, these previous studies have mostly focused on the improvement and development of building materials such as earth structures [13] and the development of new structures such as diagonal structures [14] and have largely ignored to develop a proper derivation method to evaluate the properties of the sustainable buildings’ components and materials. To this end, modern design standards [15–18] offer the possibility to carry out design approaches assisted by testing, which introduces the structural test into the design process and can determine directly the design value of components’ performance by statistical derivation based on the test data. It can be applied to the partial factor method and the design value method but has better empirical validity than them.Nevertheless, the potential of the design assisted by testing for deriving structural resistance has not been systematically studied yet. Currently, the research on the design assisted by testing mostly stays at the application level, such as the derivation of design values for new materials and complex structural properties and the assessment of the reliability of new resistance models. For example, a new design method based on the design assisted by testing for steel members and cold-formed thin-walled members was proposed in References [19, 20], where it was pointed out in References [20] that it is difficult to find a reasonable solution that takes into account the defective effect without tests. Similarly, based on the design assisted by testing, the resistance model of rolled and welded steel beams considering the defective effect using this method was established in References [21]; the resistance models and subfactors of welded steel beams, high-strength steel members, and stainless steel members were examined and improved in References [22–25], where the influence of human factors was considered in Referencs [25]; the new resistance model and reliability of bolted connections were validated in Reference [26]; a new design method for prestressed concrete inclined columns was validated in Reference [27]; the resistance models of steel-fiber concrete beams and steel-concrete composite columns were examined in Reference [26, 28]; the resistance model for unreinforced masonry walls was validated in Reference [29] with new subfactors.This paper briefly introduces the main elements and the basis for establishing the most important resistance derivation in the design assisted by testing. The characteristics and shortcomings will be analyzed from a statistical point of view based on the data collected in the literature and suggestions for improvement will be made. This paper has an important role for the improvement of the design assisted by testing.


## 2. Derivation of Design Values

### 2.1. General

Derivation of design values is a major field of interest in the safety of design methodologies assisted by testing, which derives the resistance of similar members by destruction test result. As shown in Figure1, derivation of design values is typically on the basis of variables included in the resistance (hereinafter referred to as statistical determination of a single property) only in relation to the test results or on the basis of resistance model (hereinafter referred to as statistical determination of resistance models) adopting not only test results but also probability characteristics of influence factors. The former has a feature of simple application, whereas the latter largely reduces the uncertainty and derives a better resistance design value [15, 16].Figure 1
Representation of the two methods for the design assisted by testing.Derivation of design values proposed by ISO 2394:1998 includes the interval estimation method, Bayesian procedures, and method based on the resistance model. Interval estimation and Bayesian procedures belong to statistical determination of a single property, which assume that the resistances obey a normal distribution [17]. In contrast, statistical determination of a single property proposed by EN 1990:2002 not only improves the Bayesian approach but also includes the situation where the resistances obey a log-normal distribution [15]. Similarly, only the Bayesian procedure is kept by ISO 2394:2015 [16]. For statistical determination of resistance models, ISO 2394:1998 and ISO 2394:2015 both determine the resistance design values by considering an analysis model for the structural property by the computational model uncertainty factor ηT/unknown coefficient θ [16, 17]. In contrast, EN 1990:2002 adopts a probabilistic model for resistance to derive the standard or design values of resistance by uniform guarantee rate [15].


### 2.2. Statistical Determination of a Single Property

As shown in Figure2, derivation of design values can be categorized into two: direct assessment and indirect assessment. Direct assessment derives design values by assessing a characteristic value, which may have to be adjusted by conversion factors and by applying partial factor method. Indirect assessment derives design values by direct determination of the design value from test results allowing explicitly or implicitly for conversion aspects and by applying the design value method. Bayesian procedures have both direct assessment and indirect assessment, while interval estimation methods have only indirect assessment.Figure 2
Representation of the two assessments for the derivation of design values.Indirect assessment can be defined as [15](1)Rd=ηCdRkγm,where Rk  = the characteristic value of the resistance - implied model uncertainty factor ηT; γm  = material partial factors; and ηCd  = the design value of the conversion factor ηC with the consideration of difference between reality and the test conditions.


#### 2.2.1. Interval Estimation Method

As the interval estimation method proposed by ISO 2394:1998 assumes that the resistances obey the normal distributionNμR,σR2, the characteristic value of the resistance can be expressed as [17](2)Rk=μR−z1−pkσR,where z1−pk=Φ−1s1−pk and pk  = the cumulative probability/probability of non-exceedance of characteristic value of the resistance Rk.SupposeX1,X2,⋯,Xn are resistance samples obtained from bearing capacity test. When the standard deviation of resistance σR is known, the pivotal quantity X¯−Rk/SR/n obeys the non-central t-distribution with df (degree of freedom) n−1 and non-centrality parameter z1−pkn, where X¯ is the sample mean and SR is sample standard deviation [30]. According to interval estimation method [31], Rk is defined as the characteristic value of the resistance, which can be obtained as(3)Rk=mR−tn−1,z1−pkn,CnsR,mR=1n∑i=1nxi,SR=1n−1∑i=1nxi−mR2,where x1,x2,⋯,xn  = values of resistance sample; n  = sample size; mR  = value of sample mean; sR  = value of sample standard deviation; C  = confidence level; and tn−1,z1−pkn,C  = the 1−C quantile of non-central t-distribution with df n−1 and non-centrality parameter z1−pkn.When the standard deviation of resistanceσR is known, the pivotal quantity X¯−Rk−z1−pkσR/σR/n obeys standard normal distribution N0,1. In the same way, the characteristic value Rk is defined as(4)Rk=mR−z1−pk+z1−CnσR,where z1−pk  = the C quantile of standard normal distribution.To calculate the characteristic value of the resistance, formula (5) is employed for known σR and formula (6) is employed for unknown σR [16].(5)Rk=mR−kssR,(6)Rk=mR−kσsR,where ks=tn−1,z1−pkn,C/n; kσ=z1−pk+z1−C/n.Apparently,σR is improperly replaced with sR in formula (6). It is proper, as commended by ISO 2394:1998, to give the value of the confidence level C less than 0.75 and the pk value for 0.95 without available information.It is worth noting that the value ofC and pk recommended by ISO 2394:1998 does not apply for structure in China. A value of pk less than 0.9 was considered to be appropriate, and a value of C is 0.6 was applicable to be masonry structure, 0.7 for concrete structure, 0.9 for steel construction in GB 50292-2015 [18].


#### 2.2.2. Bayesian Procedures

(1) General Principles on Reliability for Structures. ISO 2394:1998 and ISO 2394:2015 propose the Bayesian procedures to derive Rd only by the direct assessment and presume that R obeys the normal distribution NμR,σR2.The probability density function of the normal distribution is considered a conditional probability density function with unknown parameters. Its prior distribution is the Jeffreys prior with the following equation (7) [32, 33]:(7)πμR,σR∝1σR,where∝ = be proportional to.The likelihood function forR can be obtained by(8)px1,x2,…,xn|μR,σR∝1σRne−1/2∑i=1nxi−μR2/σR2∝1σRne−1/2n−1sR2+nμR−mR2/σR2.The joint posterior distribution ofμR and σR can be derived by formula (9) using Bayesian formula.(9)πμR,σR|x1,x2,…,xn∝px1,x2,…,xn|μR,σRπμR,σR∝1σRn+1e−1/2n−1sR2+nμR−mR2/σR2.The probability density function forR can be obtained by the following formula.(10)fR∝∫−∞∞∫0∞fR|μR,σRπμR,σR|x1,x2,⋯,xndσRdμR∝1+1n−1R−mRsR1+1/n2−n−1+1/2,where the pivotal quantity R−mR/sR1+1/n obeys Student’s t distribution with df n−1 [30]. Then, Rd can be obtained, based on probability definition, by(11)Rd=mR−tn−1,1−psR1+1n,where tn−1,1−p  =  1−p quantile value of t distribution with df n−1 and pd  = the guarantee rate of Rd.WithσR known, the Jeffreys prior for the unknown parameter μR is as follows [32, 33]:(12)πμR∝1.Similar to the Bayesian derivation, the pivotal quantityR−mR/σR1+1/n obeys the standard normal distribution. Rd was performed in the same way as described above:(13)Rd=mR−z1−pσR1+1n.Since theRd value, derived by the formulas (11) and (13), is under laboratory conditions, derivation of Rd proposed by ISO 2394 derived by formulas (14) and (15) includes the conversion factor η16, 17:(14)Rd=ηdmR−tvdsR1+1n,(15)tvd=tn−1,1−pσRunknown,z1−pσRknown.The above formula is direct derivation pathway, while the indirect derivation pathway is as follows:(16)Rd=ηCdRkγm=ηCdγmmR−tvdsR1+1/n,(17)tvd=tn−1,1−p,σRunknown,z1−p,σRknown.Similarly,σR is improperly replaced with sR in formulas (14–16). Furthermore, the coefficient tvd is only dependent on the sample size, n, and on the chosen guarantee rate, pk, pd, without the confidence level C.(2) Eurocode: Basis of Structural Design. The method proposed by EN is similar to ISO 2394, and derivation of Rd is as follows [15]:(18)Rd=ηdγmmR1−knVR,Rd=ηdmR1−kd,nVR,kn=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,kd,n=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,where VR  = the coefficient of variation of R; if VR is known, VR=σR/μR; if VR is unknown, VR=sR/μR; kn, kd,n depend on the sample size, n, and on the chosen guarantee rate, pk.This method of deriving theRd is similar to the approach proposed by ISO 2394 [16, 17]. However, the row “VR known” should represent “σR/μR known” but should actually represent “σR/μR,” whose derived results are completely different.As mentioned in EN 1990:2002, the resistanceR obeys the log-normal distribution LNμlnR,σlnR2. So, lnR obeys the normal distribution NμlnR,σlnR2 and the characteristic parameter estimated according to formulas (1920). With the log-normal distribution, the direct and indirect assessment of Rd becomes formulas (2122).(19)μlnR=lnμR1+VR2,(20)σlnR=ln1+VR2,(21)Rd=ηdγmexpmlnR−knslnR,(22)Rd=ηdexpmlnR−kd,nslnR,(23)mlnR=1n∑i=1nlnxi,slnR=1n−1∑i=1nlnxi−mlnR2VRunknown,ln1+VR2≈VRVRknown.It is usually assumed that the resistance follows the log-normal distribution, but the resistance of the member with simple force and controlled by a single material can be assumed to follow the normal distribution. With a normal distribution, the estimation ofRd may even appear to be negative with small sample size and large guarantee rate and variability. Fortunately, to avoid this fact, the resistance R obeying a logarithmic distribution is mentioned by EN 1990:2002.


#### 2.2.3. Evaluation Using Probabilistic Methods

Evaluation using probabilistic methods in ISO belongs to Bayesian procedures. Theoretically, the prior distribution of the Bayesian procedure can be chosen arbitrarily based on experience, which leads to the reasonableness of the Bayesian procedure without guarantee. Therefore, the joint distribution of the distribution parametersμR and σR of the method is not assumed to be Jeffreys prior but normal-inverse gamma distribution, which ensures the reasonableness of the method.The method supposes thatμR, which is the conditional probability distribution about σR2, obeys the normal distribution Nm′,σR2/N′, and σR2 obeys the inverse gamma distribution IGav′/2,v′s′2/n′, and the probability density function of σR2 is(24)fσR2x=v′s′2/2v′/2Γv′/21xv′/2+1exp−v′s′22xx≥0.Also, the mean value and coefficient of variation ofμR are determined by formulas (25) and (26). When the value of v′ is large, the mean value and coefficient of variation of σR2 are approximated according to equations (27) and (28), respectively.(25)EμR=m′,(26)VμR=s′m′n′,(27)EσR=s′,(28)VσR=12v′.If it is assumed that a series of a priori tests have been conducted before the test,m′ and s′ are equivalent to the point estimates of μR and σR, respectively, n′ is equivalent to the sample size in deriving μR, and v′ is equivalent to the degree of freedom in deriving σR. All of these parameters can be chosen independently.According to the same steps as the aforementioned Bayesian procedures, the standard value or design value of resistance can be obtained by the following formulas:(29)R=m′′−tv′′,1−ps′′1+1/n′′,n′′=n+n′,v′′=v+v′+δn′,v=n−1,δn′=0n′=01n′>0m′′n′′=n′m′+nmR,v′′s′′2+n′′m′′2=v′s′2+n′m′2+vsR2+nmR2,where tv′′,1−p  = the 1−p quantile of t-distribution with df v′′ and p  = guaranteeing rate of the characteristic or design values. When the standard deviation of the resistance σR is known, the prior distribution of μR is taken to be the normal distribution Nm′,σR2/n′ and the resistance is determined according to the following formula:(30)R=m′′−n1−pσR1+1/n′′tv∗=tv∗,1−p,σRunknown,z1−p,σRknown.


### 2.3. Derivation Methods Based on the Analysis Model

#### 2.3.1. Method in ISO 2394

For the deriving methods based on the analysis model mentioned by ISO 2394, the following basic formula is applied [16]:(31)Rd=ηdγdRxd,w,where xd  = the design value of vector of random variables; w  = the value of measurable deterministic variables; ηd  = the design value of the conversion coefficient - misnamed the design value of coefficient of the model uncertainty; γd=1/θd; and θd  = the design value of model uncertainty (also known as calculation mode uncertainty factor) that reflects the difference between the test and calculated values of the resistance needs to be derived by test results. In this standard, the model uncertainty θ is considered to be a random variable assuming a log-normal distribution and its design value θd is estimated from(32)θd=expmθ′±tvd1+1n⋅sθ′,where θi′=lnθ′i=1,2,⋯n; mθ′  = mean deviation for θi′; sθ′  = standard deviation for θi′; and θ1′,θ2′,⋯,θn′  = values of model uncertainty sample.This method is built on the relationship between the design value of resistance and the vector of random variables to obtainRd, which is formula (31). However, Since pd in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), Rd derived from formula (31) has a different guarantee rate. To clarify, for different xd, w, ηd, and θd, the guarantee rate of the design value of resistance pd is generally different or even has a large difference, which is difficult to ensure effective control of structural reliability. Therefore, this method of ISO is approximate.


#### 2.3.2. Method in EN 1990

The method in EN implied that the probability property of vector of random variables can not only deriveRd but also derive Rk. Since the vector of random variable η is difficult to determine, the method cannot derive Rd and Rk from the ideal row of VR known and VR unknown. Therefore, to solve the above problem, the method introduces a weighted average.AsR obeys the log-normal LNμlnR,σlnR2 and VR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula:(33)Rk=expmy−knsy,where my  = mean value of the logarithm of the measured resistance values; sy  = standard deviation of the logarithm of the measured resistance values; and kn=tn−1,1−pk1+1/n (VR unknown), with no information (pk=0.95).Set(34)Q=σlnR=ln1+VR2.Hence [2]:(35)my≈lnmR1+VR2=lnmR−0.5Q2,(36)sy≈ln1+VR2=Q,(37)Rk=mRexp−knQ−0.5Q2.AsVR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula [16]:(38)Rk=expmy−k∞σlnR,=mRexp−k∞Q−0.5Q2,wherek∞=z1−pk when n tends to infinity, [pk=0.95, then k∞=1.64].The probabilistic mode of the resistance was constructed as follows [16]:(39)R=ηR0⋅=ηCηTR0⋅,where R0⋅  = probabilistic structural performance function implied correction factor b; η  = the model uncertainty; and R0  = theoretical value of resistance.Supposeη and R0 both obey log-normal distribution with the following formulation:(40)lnR=lnη+lnR0⋅,Q2=Qη2+QR02,Q=σlnR=σlnη2+σlnR02,Qη=σlnη=ln1+Vη2,QR0=σlnR0=ln1+VR02,where Vη  = the coefficient of variation of η and VR0  = the coefficient of variation of R0.Because generallyVR0 is known and Vη is unknown, the derivation of Rk lies between the calculated values of formula (37) (VR known) and formula (38) (VR unknown). Consequently, the weighting factors can be defined as follows [16]:(41)αη=QηQ=ln1+Vη2ln1+VR2,αR0=QR0Q=ln1+VR02ln1+VR2,αη2+αR02=1.Obviously, the largerαη, the higher the unknown degree of VR. The larger αR0, the higher the known degree of VR. Following this weighted average method, the derivation formula for Rk is as follows:(42)Rk=mRexp−k∞αR02Q−kηαη2Q−0.5Q2=mRexp−k∞αR0QR0−knαηQη−0.5Q2,(43)Vδ=expsΔ2−1,sΔ=1n−1∑i=1nlnδi−mΔ2,mΔ=1n∑i−1nlnηi,where η1,η2,⋯ηn  = values of model uncertainty sample. If kn≥100, kn=k∞. Then, the derivation of Rk be obtained by the following formula:(44)Rk=mRexp−k∞Q−0.5Q2.By the same weighted average method, the derivation formula for the valueRd that implicitly accounts for ηC is obtained as [16](45)Rd=mRexp−k∞αR0QR0−kd,nαηQη−0.5Q2.Whenn is greater than or equal to 100, the design value of resistance was calculated according to the formula as follows.(46)Rd=mRexp−kd,∞Q−0.5Q2,where kd,n=tn−1,1−pd1+1/n; kd,∞=z1−pd; and pd=Φαβ.


## 2.1. General

Derivation of design values is a major field of interest in the safety of design methodologies assisted by testing, which derives the resistance of similar members by destruction test result. As shown in Figure1, derivation of design values is typically on the basis of variables included in the resistance (hereinafter referred to as statistical determination of a single property) only in relation to the test results or on the basis of resistance model (hereinafter referred to as statistical determination of resistance models) adopting not only test results but also probability characteristics of influence factors. The former has a feature of simple application, whereas the latter largely reduces the uncertainty and derives a better resistance design value [15, 16].Figure 1
Representation of the two methods for the design assisted by testing.Derivation of design values proposed by ISO 2394:1998 includes the interval estimation method, Bayesian procedures, and method based on the resistance model. Interval estimation and Bayesian procedures belong to statistical determination of a single property, which assume that the resistances obey a normal distribution [17]. In contrast, statistical determination of a single property proposed by EN 1990:2002 not only improves the Bayesian approach but also includes the situation where the resistances obey a log-normal distribution [15]. Similarly, only the Bayesian procedure is kept by ISO 2394:2015 [16]. For statistical determination of resistance models, ISO 2394:1998 and ISO 2394:2015 both determine the resistance design values by considering an analysis model for the structural property by the computational model uncertainty factor ηT/unknown coefficient θ [16, 17]. In contrast, EN 1990:2002 adopts a probabilistic model for resistance to derive the standard or design values of resistance by uniform guarantee rate [15].


## 2.2. Statistical Determination of a Single Property

As shown in Figure2, derivation of design values can be categorized into two: direct assessment and indirect assessment. Direct assessment derives design values by assessing a characteristic value, which may have to be adjusted by conversion factors and by applying partial factor method. Indirect assessment derives design values by direct determination of the design value from test results allowing explicitly or implicitly for conversion aspects and by applying the design value method. Bayesian procedures have both direct assessment and indirect assessment, while interval estimation methods have only indirect assessment.Figure 2
Representation of the two assessments for the derivation of design values.Indirect assessment can be defined as [15](1)Rd=ηCdRkγm,where Rk  = the characteristic value of the resistance - implied model uncertainty factor ηT; γm  = material partial factors; and ηCd  = the design value of the conversion factor ηC with the consideration of difference between reality and the test conditions.


### 2.2.1. Interval Estimation Method

As the interval estimation method proposed by ISO 2394:1998 assumes that the resistances obey the normal distributionNμR,σR2, the characteristic value of the resistance can be expressed as [17](2)Rk=μR−z1−pkσR,where z1−pk=Φ−1s1−pk and pk  = the cumulative probability/probability of non-exceedance of characteristic value of the resistance Rk.SupposeX1,X2,⋯,Xn are resistance samples obtained from bearing capacity test. When the standard deviation of resistance σR is known, the pivotal quantity X¯−Rk/SR/n obeys the non-central t-distribution with df (degree of freedom) n−1 and non-centrality parameter z1−pkn, where X¯ is the sample mean and SR is sample standard deviation [30]. According to interval estimation method [31], Rk is defined as the characteristic value of the resistance, which can be obtained as(3)Rk=mR−tn−1,z1−pkn,CnsR,mR=1n∑i=1nxi,SR=1n−1∑i=1nxi−mR2,where x1,x2,⋯,xn  = values of resistance sample; n  = sample size; mR  = value of sample mean; sR  = value of sample standard deviation; C  = confidence level; and tn−1,z1−pkn,C  = the 1−C quantile of non-central t-distribution with df n−1 and non-centrality parameter z1−pkn.When the standard deviation of resistanceσR is known, the pivotal quantity X¯−Rk−z1−pkσR/σR/n obeys standard normal distribution N0,1. In the same way, the characteristic value Rk is defined as(4)Rk=mR−z1−pk+z1−CnσR,where z1−pk  = the C quantile of standard normal distribution.To calculate the characteristic value of the resistance, formula (5) is employed for known σR and formula (6) is employed for unknown σR [16].(5)Rk=mR−kssR,(6)Rk=mR−kσsR,where ks=tn−1,z1−pkn,C/n; kσ=z1−pk+z1−C/n.Apparently,σR is improperly replaced with sR in formula (6). It is proper, as commended by ISO 2394:1998, to give the value of the confidence level C less than 0.75 and the pk value for 0.95 without available information.It is worth noting that the value ofC and pk recommended by ISO 2394:1998 does not apply for structure in China. A value of pk less than 0.9 was considered to be appropriate, and a value of C is 0.6 was applicable to be masonry structure, 0.7 for concrete structure, 0.9 for steel construction in GB 50292-2015 [18].


### 2.2.2. Bayesian Procedures

(1) General Principles on Reliability for Structures. ISO 2394:1998 and ISO 2394:2015 propose the Bayesian procedures to derive Rd only by the direct assessment and presume that R obeys the normal distribution NμR,σR2.The probability density function of the normal distribution is considered a conditional probability density function with unknown parameters. Its prior distribution is the Jeffreys prior with the following equation (7) [32, 33]:(7)πμR,σR∝1σR,where∝ = be proportional to.The likelihood function forR can be obtained by(8)px1,x2,…,xn|μR,σR∝1σRne−1/2∑i=1nxi−μR2/σR2∝1σRne−1/2n−1sR2+nμR−mR2/σR2.The joint posterior distribution ofμR and σR can be derived by formula (9) using Bayesian formula.(9)πμR,σR|x1,x2,…,xn∝px1,x2,…,xn|μR,σRπμR,σR∝1σRn+1e−1/2n−1sR2+nμR−mR2/σR2.The probability density function forR can be obtained by the following formula.(10)fR∝∫−∞∞∫0∞fR|μR,σRπμR,σR|x1,x2,⋯,xndσRdμR∝1+1n−1R−mRsR1+1/n2−n−1+1/2,where the pivotal quantity R−mR/sR1+1/n obeys Student’s t distribution with df n−1 [30]. Then, Rd can be obtained, based on probability definition, by(11)Rd=mR−tn−1,1−psR1+1n,where tn−1,1−p  =  1−p quantile value of t distribution with df n−1 and pd  = the guarantee rate of Rd.WithσR known, the Jeffreys prior for the unknown parameter μR is as follows [32, 33]:(12)πμR∝1.Similar to the Bayesian derivation, the pivotal quantityR−mR/σR1+1/n obeys the standard normal distribution. Rd was performed in the same way as described above:(13)Rd=mR−z1−pσR1+1n.Since theRd value, derived by the formulas (11) and (13), is under laboratory conditions, derivation of Rd proposed by ISO 2394 derived by formulas (14) and (15) includes the conversion factor η16, 17:(14)Rd=ηdmR−tvdsR1+1n,(15)tvd=tn−1,1−pσRunknown,z1−pσRknown.The above formula is direct derivation pathway, while the indirect derivation pathway is as follows:(16)Rd=ηCdRkγm=ηCdγmmR−tvdsR1+1/n,(17)tvd=tn−1,1−p,σRunknown,z1−p,σRknown.Similarly,σR is improperly replaced with sR in formulas (14–16). Furthermore, the coefficient tvd is only dependent on the sample size, n, and on the chosen guarantee rate, pk, pd, without the confidence level C.(2) Eurocode: Basis of Structural Design. The method proposed by EN is similar to ISO 2394, and derivation of Rd is as follows [15]:(18)Rd=ηdγmmR1−knVR,Rd=ηdmR1−kd,nVR,kn=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,kd,n=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,where VR  = the coefficient of variation of R; if VR is known, VR=σR/μR; if VR is unknown, VR=sR/μR; kn, kd,n depend on the sample size, n, and on the chosen guarantee rate, pk.This method of deriving theRd is similar to the approach proposed by ISO 2394 [16, 17]. However, the row “VR known” should represent “σR/μR known” but should actually represent “σR/μR,” whose derived results are completely different.As mentioned in EN 1990:2002, the resistanceR obeys the log-normal distribution LNμlnR,σlnR2. So, lnR obeys the normal distribution NμlnR,σlnR2 and the characteristic parameter estimated according to formulas (1920). With the log-normal distribution, the direct and indirect assessment of Rd becomes formulas (2122).(19)μlnR=lnμR1+VR2,(20)σlnR=ln1+VR2,(21)Rd=ηdγmexpmlnR−knslnR,(22)Rd=ηdexpmlnR−kd,nslnR,(23)mlnR=1n∑i=1nlnxi,slnR=1n−1∑i=1nlnxi−mlnR2VRunknown,ln1+VR2≈VRVRknown.It is usually assumed that the resistance follows the log-normal distribution, but the resistance of the member with simple force and controlled by a single material can be assumed to follow the normal distribution. With a normal distribution, the estimation ofRd may even appear to be negative with small sample size and large guarantee rate and variability. Fortunately, to avoid this fact, the resistance R obeying a logarithmic distribution is mentioned by EN 1990:2002.


### 2.2.3. Evaluation Using Probabilistic Methods

Evaluation using probabilistic methods in ISO belongs to Bayesian procedures. Theoretically, the prior distribution of the Bayesian procedure can be chosen arbitrarily based on experience, which leads to the reasonableness of the Bayesian procedure without guarantee. Therefore, the joint distribution of the distribution parametersμR and σR of the method is not assumed to be Jeffreys prior but normal-inverse gamma distribution, which ensures the reasonableness of the method.The method supposes thatμR, which is the conditional probability distribution about σR2, obeys the normal distribution Nm′,σR2/N′, and σR2 obeys the inverse gamma distribution IGav′/2,v′s′2/n′, and the probability density function of σR2 is(24)fσR2x=v′s′2/2v′/2Γv′/21xv′/2+1exp−v′s′22xx≥0.Also, the mean value and coefficient of variation ofμR are determined by formulas (25) and (26). When the value of v′ is large, the mean value and coefficient of variation of σR2 are approximated according to equations (27) and (28), respectively.(25)EμR=m′,(26)VμR=s′m′n′,(27)EσR=s′,(28)VσR=12v′.If it is assumed that a series of a priori tests have been conducted before the test,m′ and s′ are equivalent to the point estimates of μR and σR, respectively, n′ is equivalent to the sample size in deriving μR, and v′ is equivalent to the degree of freedom in deriving σR. All of these parameters can be chosen independently.According to the same steps as the aforementioned Bayesian procedures, the standard value or design value of resistance can be obtained by the following formulas:(29)R=m′′−tv′′,1−ps′′1+1/n′′,n′′=n+n′,v′′=v+v′+δn′,v=n−1,δn′=0n′=01n′>0m′′n′′=n′m′+nmR,v′′s′′2+n′′m′′2=v′s′2+n′m′2+vsR2+nmR2,where tv′′,1−p  = the 1−p quantile of t-distribution with df v′′ and p  = guaranteeing rate of the characteristic or design values. When the standard deviation of the resistance σR is known, the prior distribution of μR is taken to be the normal distribution Nm′,σR2/n′ and the resistance is determined according to the following formula:(30)R=m′′−n1−pσR1+1/n′′tv∗=tv∗,1−p,σRunknown,z1−p,σRknown.


## 2.2.1. Interval Estimation Method

As the interval estimation method proposed by ISO 2394:1998 assumes that the resistances obey the normal distributionNμR,σR2, the characteristic value of the resistance can be expressed as [17](2)Rk=μR−z1−pkσR,where z1−pk=Φ−1s1−pk and pk  = the cumulative probability/probability of non-exceedance of characteristic value of the resistance Rk.SupposeX1,X2,⋯,Xn are resistance samples obtained from bearing capacity test. When the standard deviation of resistance σR is known, the pivotal quantity X¯−Rk/SR/n obeys the non-central t-distribution with df (degree of freedom) n−1 and non-centrality parameter z1−pkn, where X¯ is the sample mean and SR is sample standard deviation [30]. According to interval estimation method [31], Rk is defined as the characteristic value of the resistance, which can be obtained as(3)Rk=mR−tn−1,z1−pkn,CnsR,mR=1n∑i=1nxi,SR=1n−1∑i=1nxi−mR2,where x1,x2,⋯,xn  = values of resistance sample; n  = sample size; mR  = value of sample mean; sR  = value of sample standard deviation; C  = confidence level; and tn−1,z1−pkn,C  = the 1−C quantile of non-central t-distribution with df n−1 and non-centrality parameter z1−pkn.When the standard deviation of resistanceσR is known, the pivotal quantity X¯−Rk−z1−pkσR/σR/n obeys standard normal distribution N0,1. In the same way, the characteristic value Rk is defined as(4)Rk=mR−z1−pk+z1−CnσR,where z1−pk  = the C quantile of standard normal distribution.To calculate the characteristic value of the resistance, formula (5) is employed for known σR and formula (6) is employed for unknown σR [16].(5)Rk=mR−kssR,(6)Rk=mR−kσsR,where ks=tn−1,z1−pkn,C/n; kσ=z1−pk+z1−C/n.Apparently,σR is improperly replaced with sR in formula (6). It is proper, as commended by ISO 2394:1998, to give the value of the confidence level C less than 0.75 and the pk value for 0.95 without available information.It is worth noting that the value ofC and pk recommended by ISO 2394:1998 does not apply for structure in China. A value of pk less than 0.9 was considered to be appropriate, and a value of C is 0.6 was applicable to be masonry structure, 0.7 for concrete structure, 0.9 for steel construction in GB 50292-2015 [18].


## 2.2.2. Bayesian Procedures

(1) General Principles on Reliability for Structures. ISO 2394:1998 and ISO 2394:2015 propose the Bayesian procedures to derive Rd only by the direct assessment and presume that R obeys the normal distribution NμR,σR2.The probability density function of the normal distribution is considered a conditional probability density function with unknown parameters. Its prior distribution is the Jeffreys prior with the following equation (7) [32, 33]:(7)πμR,σR∝1σR,where∝ = be proportional to.The likelihood function forR can be obtained by(8)px1,x2,…,xn|μR,σR∝1σRne−1/2∑i=1nxi−μR2/σR2∝1σRne−1/2n−1sR2+nμR−mR2/σR2.The joint posterior distribution ofμR and σR can be derived by formula (9) using Bayesian formula.(9)πμR,σR|x1,x2,…,xn∝px1,x2,…,xn|μR,σRπμR,σR∝1σRn+1e−1/2n−1sR2+nμR−mR2/σR2.The probability density function forR can be obtained by the following formula.(10)fR∝∫−∞∞∫0∞fR|μR,σRπμR,σR|x1,x2,⋯,xndσRdμR∝1+1n−1R−mRsR1+1/n2−n−1+1/2,where the pivotal quantity R−mR/sR1+1/n obeys Student’s t distribution with df n−1 [30]. Then, Rd can be obtained, based on probability definition, by(11)Rd=mR−tn−1,1−psR1+1n,where tn−1,1−p  =  1−p quantile value of t distribution with df n−1 and pd  = the guarantee rate of Rd.WithσR known, the Jeffreys prior for the unknown parameter μR is as follows [32, 33]:(12)πμR∝1.Similar to the Bayesian derivation, the pivotal quantityR−mR/σR1+1/n obeys the standard normal distribution. Rd was performed in the same way as described above:(13)Rd=mR−z1−pσR1+1n.Since theRd value, derived by the formulas (11) and (13), is under laboratory conditions, derivation of Rd proposed by ISO 2394 derived by formulas (14) and (15) includes the conversion factor η16, 17:(14)Rd=ηdmR−tvdsR1+1n,(15)tvd=tn−1,1−pσRunknown,z1−pσRknown.The above formula is direct derivation pathway, while the indirect derivation pathway is as follows:(16)Rd=ηCdRkγm=ηCdγmmR−tvdsR1+1/n,(17)tvd=tn−1,1−p,σRunknown,z1−p,σRknown.Similarly,σR is improperly replaced with sR in formulas (14–16). Furthermore, the coefficient tvd is only dependent on the sample size, n, and on the chosen guarantee rate, pk, pd, without the confidence level C.(2) Eurocode: Basis of Structural Design. The method proposed by EN is similar to ISO 2394, and derivation of Rd is as follows [15]:(18)Rd=ηdγmmR1−knVR,Rd=ηdmR1−kd,nVR,kn=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,kd,n=tn−1,1−p1+1nVRunknown,z1−p1+1nVRknown,where VR  = the coefficient of variation of R; if VR is known, VR=σR/μR; if VR is unknown, VR=sR/μR; kn, kd,n depend on the sample size, n, and on the chosen guarantee rate, pk.This method of deriving theRd is similar to the approach proposed by ISO 2394 [16, 17]. However, the row “VR known” should represent “σR/μR known” but should actually represent “σR/μR,” whose derived results are completely different.As mentioned in EN 1990:2002, the resistanceR obeys the log-normal distribution LNμlnR,σlnR2. So, lnR obeys the normal distribution NμlnR,σlnR2 and the characteristic parameter estimated according to formulas (1920). With the log-normal distribution, the direct and indirect assessment of Rd becomes formulas (2122).(19)μlnR=lnμR1+VR2,(20)σlnR=ln1+VR2,(21)Rd=ηdγmexpmlnR−knslnR,(22)Rd=ηdexpmlnR−kd,nslnR,(23)mlnR=1n∑i=1nlnxi,slnR=1n−1∑i=1nlnxi−mlnR2VRunknown,ln1+VR2≈VRVRknown.It is usually assumed that the resistance follows the log-normal distribution, but the resistance of the member with simple force and controlled by a single material can be assumed to follow the normal distribution. With a normal distribution, the estimation ofRd may even appear to be negative with small sample size and large guarantee rate and variability. Fortunately, to avoid this fact, the resistance R obeying a logarithmic distribution is mentioned by EN 1990:2002.


## 2.2.3. Evaluation Using Probabilistic Methods

Evaluation using probabilistic methods in ISO belongs to Bayesian procedures. Theoretically, the prior distribution of the Bayesian procedure can be chosen arbitrarily based on experience, which leads to the reasonableness of the Bayesian procedure without guarantee. Therefore, the joint distribution of the distribution parametersμR and σR of the method is not assumed to be Jeffreys prior but normal-inverse gamma distribution, which ensures the reasonableness of the method.The method supposes thatμR, which is the conditional probability distribution about σR2, obeys the normal distribution Nm′,σR2/N′, and σR2 obeys the inverse gamma distribution IGav′/2,v′s′2/n′, and the probability density function of σR2 is(24)fσR2x=v′s′2/2v′/2Γv′/21xv′/2+1exp−v′s′22xx≥0.Also, the mean value and coefficient of variation ofμR are determined by formulas (25) and (26). When the value of v′ is large, the mean value and coefficient of variation of σR2 are approximated according to equations (27) and (28), respectively.(25)EμR=m′,(26)VμR=s′m′n′,(27)EσR=s′,(28)VσR=12v′.If it is assumed that a series of a priori tests have been conducted before the test,m′ and s′ are equivalent to the point estimates of μR and σR, respectively, n′ is equivalent to the sample size in deriving μR, and v′ is equivalent to the degree of freedom in deriving σR. All of these parameters can be chosen independently.According to the same steps as the aforementioned Bayesian procedures, the standard value or design value of resistance can be obtained by the following formulas:(29)R=m′′−tv′′,1−ps′′1+1/n′′,n′′=n+n′,v′′=v+v′+δn′,v=n−1,δn′=0n′=01n′>0m′′n′′=n′m′+nmR,v′′s′′2+n′′m′′2=v′s′2+n′m′2+vsR2+nmR2,where tv′′,1−p  = the 1−p quantile of t-distribution with df v′′ and p  = guaranteeing rate of the characteristic or design values. When the standard deviation of the resistance σR is known, the prior distribution of μR is taken to be the normal distribution Nm′,σR2/n′ and the resistance is determined according to the following formula:(30)R=m′′−n1−pσR1+1/n′′tv∗=tv∗,1−p,σRunknown,z1−p,σRknown.


## 2.3. Derivation Methods Based on the Analysis Model

### 2.3.1. Method in ISO 2394

For the deriving methods based on the analysis model mentioned by ISO 2394, the following basic formula is applied [16]:(31)Rd=ηdγdRxd,w,where xd  = the design value of vector of random variables; w  = the value of measurable deterministic variables; ηd  = the design value of the conversion coefficient - misnamed the design value of coefficient of the model uncertainty; γd=1/θd; and θd  = the design value of model uncertainty (also known as calculation mode uncertainty factor) that reflects the difference between the test and calculated values of the resistance needs to be derived by test results. In this standard, the model uncertainty θ is considered to be a random variable assuming a log-normal distribution and its design value θd is estimated from(32)θd=expmθ′±tvd1+1n⋅sθ′,where θi′=lnθ′i=1,2,⋯n; mθ′  = mean deviation for θi′; sθ′  = standard deviation for θi′; and θ1′,θ2′,⋯,θn′  = values of model uncertainty sample.This method is built on the relationship between the design value of resistance and the vector of random variables to obtainRd, which is formula (31). However, Since pd in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), Rd derived from formula (31) has a different guarantee rate. To clarify, for different xd, w, ηd, and θd, the guarantee rate of the design value of resistance pd is generally different or even has a large difference, which is difficult to ensure effective control of structural reliability. Therefore, this method of ISO is approximate.


### 2.3.2. Method in EN 1990

The method in EN implied that the probability property of vector of random variables can not only deriveRd but also derive Rk. Since the vector of random variable η is difficult to determine, the method cannot derive Rd and Rk from the ideal row of VR known and VR unknown. Therefore, to solve the above problem, the method introduces a weighted average.AsR obeys the log-normal LNμlnR,σlnR2 and VR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula:(33)Rk=expmy−knsy,where my  = mean value of the logarithm of the measured resistance values; sy  = standard deviation of the logarithm of the measured resistance values; and kn=tn−1,1−pk1+1/n (VR unknown), with no information (pk=0.95).Set(34)Q=σlnR=ln1+VR2.Hence [2]:(35)my≈lnmR1+VR2=lnmR−0.5Q2,(36)sy≈ln1+VR2=Q,(37)Rk=mRexp−knQ−0.5Q2.AsVR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula [16]:(38)Rk=expmy−k∞σlnR,=mRexp−k∞Q−0.5Q2,wherek∞=z1−pk when n tends to infinity, [pk=0.95, then k∞=1.64].The probabilistic mode of the resistance was constructed as follows [16]:(39)R=ηR0⋅=ηCηTR0⋅,where R0⋅  = probabilistic structural performance function implied correction factor b; η  = the model uncertainty; and R0  = theoretical value of resistance.Supposeη and R0 both obey log-normal distribution with the following formulation:(40)lnR=lnη+lnR0⋅,Q2=Qη2+QR02,Q=σlnR=σlnη2+σlnR02,Qη=σlnη=ln1+Vη2,QR0=σlnR0=ln1+VR02,where Vη  = the coefficient of variation of η and VR0  = the coefficient of variation of R0.Because generallyVR0 is known and Vη is unknown, the derivation of Rk lies between the calculated values of formula (37) (VR known) and formula (38) (VR unknown). Consequently, the weighting factors can be defined as follows [16]:(41)αη=QηQ=ln1+Vη2ln1+VR2,αR0=QR0Q=ln1+VR02ln1+VR2,αη2+αR02=1.Obviously, the largerαη, the higher the unknown degree of VR. The larger αR0, the higher the known degree of VR. Following this weighted average method, the derivation formula for Rk is as follows:(42)Rk=mRexp−k∞αR02Q−kηαη2Q−0.5Q2=mRexp−k∞αR0QR0−knαηQη−0.5Q2,(43)Vδ=expsΔ2−1,sΔ=1n−1∑i=1nlnδi−mΔ2,mΔ=1n∑i−1nlnηi,where η1,η2,⋯ηn  = values of model uncertainty sample. If kn≥100, kn=k∞. Then, the derivation of Rk be obtained by the following formula:(44)Rk=mRexp−k∞Q−0.5Q2.By the same weighted average method, the derivation formula for the valueRd that implicitly accounts for ηC is obtained as [16](45)Rd=mRexp−k∞αR0QR0−kd,nαηQη−0.5Q2.Whenn is greater than or equal to 100, the design value of resistance was calculated according to the formula as follows.(46)Rd=mRexp−kd,∞Q−0.5Q2,where kd,n=tn−1,1−pd1+1/n; kd,∞=z1−pd; and pd=Φαβ.


## 2.3.1. Method in ISO 2394

For the deriving methods based on the analysis model mentioned by ISO 2394, the following basic formula is applied [16]:(31)Rd=ηdγdRxd,w,where xd  = the design value of vector of random variables; w  = the value of measurable deterministic variables; ηd  = the design value of the conversion coefficient - misnamed the design value of coefficient of the model uncertainty; γd=1/θd; and θd  = the design value of model uncertainty (also known as calculation mode uncertainty factor) that reflects the difference between the test and calculated values of the resistance needs to be derived by test results. In this standard, the model uncertainty θ is considered to be a random variable assuming a log-normal distribution and its design value θd is estimated from(32)θd=expmθ′±tvd1+1n⋅sθ′,where θi′=lnθ′i=1,2,⋯n; mθ′  = mean deviation for θi′; sθ′  = standard deviation for θi′; and θ1′,θ2′,⋯,θn′  = values of model uncertainty sample.This method is built on the relationship between the design value of resistance and the vector of random variables to obtainRd, which is formula (31). However, Since pd in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), Rd derived from formula (31) has a different guarantee rate. To clarify, for different xd, w, ηd, and θd, the guarantee rate of the design value of resistance pd is generally different or even has a large difference, which is difficult to ensure effective control of structural reliability. Therefore, this method of ISO is approximate.


## 2.3.2. Method in EN 1990

The method in EN implied that the probability property of vector of random variables can not only deriveRd but also derive Rk. Since the vector of random variable η is difficult to determine, the method cannot derive Rd and Rk from the ideal row of VR known and VR unknown. Therefore, to solve the above problem, the method introduces a weighted average.AsR obeys the log-normal LNμlnR,σlnR2 and VR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula:(33)Rk=expmy−knsy,where my  = mean value of the logarithm of the measured resistance values; sy  = standard deviation of the logarithm of the measured resistance values; and kn=tn−1,1−pk1+1/n (VR unknown), with no information (pk=0.95).Set(34)Q=σlnR=ln1+VR2.Hence [2]:(35)my≈lnmR1+VR2=lnmR−0.5Q2,(36)sy≈ln1+VR2=Q,(37)Rk=mRexp−knQ−0.5Q2.AsVR or σlnR2 is unknown, the Bayesian procedure mentioned by EN 1990 derives Rk according to the following formula [16]:(38)Rk=expmy−k∞σlnR,=mRexp−k∞Q−0.5Q2,wherek∞=z1−pk when n tends to infinity, [pk=0.95, then k∞=1.64].The probabilistic mode of the resistance was constructed as follows [16]:(39)R=ηR0⋅=ηCηTR0⋅,where R0⋅  = probabilistic structural performance function implied correction factor b; η  = the model uncertainty; and R0  = theoretical value of resistance.Supposeη and R0 both obey log-normal distribution with the following formulation:(40)lnR=lnη+lnR0⋅,Q2=Qη2+QR02,Q=σlnR=σlnη2+σlnR02,Qη=σlnη=ln1+Vη2,QR0=σlnR0=ln1+VR02,where Vη  = the coefficient of variation of η and VR0  = the coefficient of variation of R0.Because generallyVR0 is known and Vη is unknown, the derivation of Rk lies between the calculated values of formula (37) (VR known) and formula (38) (VR unknown). Consequently, the weighting factors can be defined as follows [16]:(41)αη=QηQ=ln1+Vη2ln1+VR2,αR0=QR0Q=ln1+VR02ln1+VR2,αη2+αR02=1.Obviously, the largerαη, the higher the unknown degree of VR. The larger αR0, the higher the known degree of VR. Following this weighted average method, the derivation formula for Rk is as follows:(42)Rk=mRexp−k∞αR02Q−kηαη2Q−0.5Q2=mRexp−k∞αR0QR0−knαηQη−0.5Q2,(43)Vδ=expsΔ2−1,sΔ=1n−1∑i=1nlnδi−mΔ2,mΔ=1n∑i−1nlnηi,where η1,η2,⋯ηn  = values of model uncertainty sample. If kn≥100, kn=k∞. Then, the derivation of Rk be obtained by the following formula:(44)Rk=mRexp−k∞Q−0.5Q2.By the same weighted average method, the derivation formula for the valueRd that implicitly accounts for ηC is obtained as [16](45)Rd=mRexp−k∞αR0QR0−kd,nαηQη−0.5Q2.Whenn is greater than or equal to 100, the design value of resistance was calculated according to the formula as follows.(46)Rd=mRexp−kd,∞Q−0.5Q2,where kd,n=tn−1,1−pd1+1/n; kd,∞=z1−pd; and pd=Φαβ.


## 3. Comparison and Analysis

### 3.1. Interval Estimation Methods and Bayesian Procedures

Since evaluation using probabilistic methods belongs to Bayesian procedures, only the derivation results of knot interval estimation methods and Bayesian procedures will be analyzed. The designer can derive the design value of the resistance by Bayesian procedures and interval estimation method. However, these two derivation methods could not guarantee that the results are all consistent. Here, we take derivingRd by indirect assessment with σR unknown as an example to illustrate differences in results.In order to better reflect the difference between the two methods, design expressions in dimensionless form are used in this paper. The accuracy of the dimensionless form for the reliability analysis results of engineering structures has been used in the normal service limit state analysis [34, 35].Therefore, this paper introduced the dimensionless variable, Rχ, defined by formulas (49) and (50).Table1 presents details on the statistical parameters of resistance of structural members, where p is calculated from formula (47), and other parameters are obtained from GB 50153-2008 [18].(47)p=1−Φln1+δR2/χRln1+δR2,(48)Rχ=RkmR=1χR,(49)Rχ=1−kσVR,(50)Rχ=ηd1−kd,nVR.Table 1
Statistical parameters of resistance of various structural members.
Types of structural membersStrained conditionχRVRpRχθVθpθReinforced concrete structural membersAxial tension1.1000.1000.8171.000.040.492Axial compression (short column)1.480.1700.9871.000.050.490Small eccentric compression (short column)1.300.1500.9541.000.050.490Large eccentric compression (short column)1.160.1300.8601.000.050.490Bending1.130.1000.8801.000.040.492Shearing1.240.1900.8531.000.0150.470Figure3 presents the differences between the interval estimation method and Bayesian procedures for the derivation of values with different structure members with unknown σR. Figure 4 presents the differences with known σR. From Figures 3 and 4, it can be seen that(1)
The characterized values of resistance derived by the two methods are different, and the difference is significant when the number of testsn is small and p value is large.(2)
Compared to interval estimation, Bayesian procedures tend to be conservative with smaller sample sizen and larger guarantee rate p and adventurous in other cases.(3)
With a small number of testsn, as in Figure 3(c), the characterized values of resistance derived from the normal distribution may be negative, which is more likely to occur in the Bayesian procedures [36].(4)
With a known coefficient of variationθ, the derived results are better than unknown coefficients of variation, especially for a small number of tests n.Figure 3
Deriving results when the coefficient of variation ofθ is unknown. (a) Axial tension. (b) Axial compression (short column). (c) Small eccentric compression (short column). (d) Large eccentric compression (short column). (e) Bending. (f) Shearing. Gray denotes the standard curve, blue represents the derivation curve by the interval estimation method, and green represents the derivation curve by Bayesian procedures.
(a)(b)(c)(d)(e)(f)Figure 4
Deriving results when the coefficient of variation ofθ is known. (a) Axial tension. (b) Axial compression (short column). (c) Small eccentric compression (short column). (d) Large eccentric compression (short column). (e) Bending. (f) Shearing. Gray denotes the standard curve, blue represents the derivation curve by the interval estimation method, and green represents the derivation curve by Bayesian procedures.
(a)(b)(c)(d)(e)(f)Differences can lead to doubts in the choice of methods in the derived formulas and results between interval estimation and Bayesian methods, which is one of the reasons why only Bayesian procedures were chosen in EN 1990:2002 and ISO 2394:2015. The Bayesian procedures not only consider all possible values and possibilities of the distribution parameters but also can determine the probability distribution of the derived quantities, which is conducive to further probability analysis and suitable for establishing probabilistic models of structural performance. However, the Bayesian method does not set the confidence level in the derivation. To elaborate, the Bayesian procedures cannot explicitly determine the confidence level on the derived results for the derivation of the design value of the adversarial force. Although certain confidence levels are implied in the Bayesian procedure derived results, they vary with the sample size. Moreover, the implied confidence level is not suitable for the derivation of the standard and design values of the adversarial force because it is difficult to conform to the value law corresponding to the variability of random variables.


### 3.2. ISO 2394 and EN 1990

Since there are few research data about concrete axially tensioned members, the analysis of axially tensioned members is not considered. The values of model uncertainty sampleθ for five types of structural members and additional information on data collection and references are available in Supplementary Material A.Similar to the formula in Section3.1, this section transforms the formulas (31) and (45) as follows (formulas (51) and (52)). If the resistance function of the basic variable is a product function, then VR0 is calculated by formula (53). If the resistance function is a more complex functional form, then VR0 is calculated by formula (54). Table 2 gives the GB 50153-2008 recommended resistance functions for various structural members of concrete structures with their corresponding VR0 values [18].(51)Rχ=θkχRRxk,w,(52)Rχ=exp−k∞αR0QR0−knαηQη−0.5Q2,(53)VR02=∑i=1jVxi2,(54)VR02=VARgR0X¯gR02X¯m≅1gR02X¯m×∑i=1j∂gR0∂Xiσi2.Table 2
Resistance function of various structural members.
Types of structural membersStrained conditionResistance analysis modelVR0Reinforced concrete structural membersAxial tensionRk=fyAs0.0948Axial compression (short column)Rk=0.9ϕfcA+fy′As′=0.9ϕbhfc+fy′ρs′0.0836Large eccentric compression (short column)Rk=α1fcbx+fy′As′+∑NsiRk=α1fcbxh0−x/2+fy′As′h0−as′+∑Msi/e0.0911BendingRk=α1fcbxh0−x/2+fy′As′h0−as′0.0548ShearingRk=Vcs+Vp+0.8fyAsbsinαs+0.8fpyApbsinαpVcs=αcvftbh0+fyvAsv/sh0Vp=0.05Np00.0551The method proposed by ISO 2394 uses the analysis model in design, while the method proposed by EN 1990 uses the probabilistic model of resistance, which is one of the reasons for the difference in their derived results.Figure5 presents the differences between the ISO 2394 method and EN 1990 method for the derivation of values with different structure members with unknown σR. From Figure 5, it can be seen that(1)
It shows that the derived results using the EN 1990:2002 method are lower than those using the ISO 2394:2015 method with unknownσR. The method of EN 1990:2002 is too conservative because the derived results are low and even decrease when the number is low, while ISO 2394 derived high.(2)
The higher the variability of the resistance influencing factors, the lower the derived value of the design resistance, as shown in Figure5(b).(3)
In most cases, the ISO method can derive higher resistance values, but it can only be applied to structural elements with known resistance analysis models.Figure 5
Deriving results when the coefficient of variation ofθ is unknown. (a) Axial compression (short column). (b) Small eccentric compression (short column). (c) Large eccentric compression (short column). (d) Bending. (e) Shearing. Gray denotes the standard curve, blue represents the derivation curve by ISO 2394, and green represents the derivation curve by EN 1990.
(a)(b)(c)(d)(e)The ISO 2394 method is built on the relationship between the design value of resistance and the vector of random variables to obtainRd, which is formula (31). However, since pd in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), Rd derived from formula (31) has a different guarantee rate. To clarify, for different xd, w, ηd, and θd, the guarantee rate of the design value of resistance pd is generally different or even has a large difference, which is difficult to ensure effective control of structural reliability. Therefore, this method of ISO is approximate.The method mentioned in EN 1990 is empirical in character because the derivation ofRk and Rd is artificially weighted and averaged. Besides, the method has the following shortcomings in the derivation process considered.(1)
Formulas (37) and (38) are both established using the Bayesian procedures without setting the confidence level, which is not proper for the derivation of Rd.(2)
Formula (38) incorrectly adopts k∞=z1−pk, but in fact k∞=z1−pk1+1/n. When n is large, it can be considered that k∞=z1−pk, which does not apply to other cases.(3)
Formula (35) does not hold between my and mR. my=μlnR and mR=μR, and μlnR has the following relationship with μR, which does not apply to the small sample condition:(55)μlnR=lnμR1+VR2.(4)
Since formulas (3738) correspond to the condition that VR is completely unknown and completely known, respectively, Q in the two formulas should have been completely unknown and completely known, respectively. However, the mean value of Q in the method mentioned by EN 1900 is established according to Vη in formula (42), which does not fit the case that VR is known and unknown. In summary, although the approximation in the method is very necessary, its theory is not rigorous.


## 3.1. Interval Estimation Methods and Bayesian Procedures

Since evaluation using probabilistic methods belongs to Bayesian procedures, only the derivation results of knot interval estimation methods and Bayesian procedures will be analyzed. The designer can derive the design value of the resistance by Bayesian procedures and interval estimation method. However, these two derivation methods could not guarantee that the results are all consistent. Here, we take derivingRd by indirect assessment with σR unknown as an example to illustrate differences in results.In order to better reflect the difference between the two methods, design expressions in dimensionless form are used in this paper. The accuracy of the dimensionless form for the reliability analysis results of engineering structures has been used in the normal service limit state analysis [34, 35].Therefore, this paper introduced the dimensionless variable, Rχ, defined by formulas (49) and (50).Table1 presents details on the statistical parameters of resistance of structural members, where p is calculated from formula (47), and other parameters are obtained from GB 50153-2008 [18].(47)p=1−Φln1+δR2/χRln1+δR2,(48)Rχ=RkmR=1χR,(49)Rχ=1−kσVR,(50)Rχ=ηd1−kd,nVR.Table 1
Statistical parameters of resistance of various structural members.
Types of structural membersStrained conditionχRVRpRχθVθpθReinforced concrete structural membersAxial tension1.1000.1000.8171.000.040.492Axial compression (short column)1.480.1700.9871.000.050.490Small eccentric compression (short column)1.300.1500.9541.000.050.490Large eccentric compression (short column)1.160.1300.8601.000.050.490Bending1.130.1000.8801.000.040.492Shearing1.240.1900.8531.000.0150.470Figure3 presents the differences between the interval estimation method and Bayesian procedures for the derivation of values with different structure members with unknown σR. Figure 4 presents the differences with known σR. From Figures 3 and 4, it can be seen that(1)
The characterized values of resistance derived by the two methods are different, and the difference is significant when the number of testsn is small and p value is large.(2)
Compared to interval estimation, Bayesian procedures tend to be conservative with smaller sample sizen and larger guarantee rate p and adventurous in other cases.(3)
With a small number of testsn, as in Figure 3(c), the characterized values of resistance derived from the normal distribution may be negative, which is more likely to occur in the Bayesian procedures [36].(4)
With a known coefficient of variationθ, the derived results are better than unknown coefficients of variation, especially for a small number of tests n.Figure 3
Deriving results when the coefficient of variation ofθ is unknown. (a) Axial tension. (b) Axial compression (short column). (c) Small eccentric compression (short column). (d) Large eccentric compression (short column). (e) Bending. (f) Shearing. Gray denotes the standard curve, blue represents the derivation curve by the interval estimation method, and green represents the derivation curve by Bayesian procedures.
(a)(b)(c)(d)(e)(f)Figure 4
Deriving results when the coefficient of variation ofθ is known. (a) Axial tension. (b) Axial compression (short column). (c) Small eccentric compression (short column). (d) Large eccentric compression (short column). (e) Bending. (f) Shearing. Gray denotes the standard curve, blue represents the derivation curve by the interval estimation method, and green represents the derivation curve by Bayesian procedures.
(a)(b)(c)(d)(e)(f)Differences can lead to doubts in the choice of methods in the derived formulas and results between interval estimation and Bayesian methods, which is one of the reasons why only Bayesian procedures were chosen in EN 1990:2002 and ISO 2394:2015. The Bayesian procedures not only consider all possible values and possibilities of the distribution parameters but also can determine the probability distribution of the derived quantities, which is conducive to further probability analysis and suitable for establishing probabilistic models of structural performance. However, the Bayesian method does not set the confidence level in the derivation. To elaborate, the Bayesian procedures cannot explicitly determine the confidence level on the derived results for the derivation of the design value of the adversarial force. Although certain confidence levels are implied in the Bayesian procedure derived results, they vary with the sample size. Moreover, the implied confidence level is not suitable for the derivation of the standard and design values of the adversarial force because it is difficult to conform to the value law corresponding to the variability of random variables.


## 3.2. ISO 2394 and EN 1990

Since there are few research data about concrete axially tensioned members, the analysis of axially tensioned members is not considered. The values of model uncertainty sampleθ for five types of structural members and additional information on data collection and references are available in Supplementary Material A.Similar to the formula in Section3.1, this section transforms the formulas (31) and (45) as follows (formulas (51) and (52)). If the resistance function of the basic variable is a product function, then VR0 is calculated by formula (53). If the resistance function is a more complex functional form, then VR0 is calculated by formula (54). Table 2 gives the GB 50153-2008 recommended resistance functions for various structural members of concrete structures with their corresponding VR0 values [18].(51)Rχ=θkχRRxk,w,(52)Rχ=exp−k∞αR0QR0−knαηQη−0.5Q2,(53)VR02=∑i=1jVxi2,(54)VR02=VARgR0X¯gR02X¯m≅1gR02X¯m×∑i=1j∂gR0∂Xiσi2.Table 2
Resistance function of various structural members.
Types of structural membersStrained conditionResistance analysis modelVR0Reinforced concrete structural membersAxial tensionRk=fyAs0.0948Axial compression (short column)Rk=0.9ϕfcA+fy′As′=0.9ϕbhfc+fy′ρs′0.0836Large eccentric compression (short column)Rk=α1fcbx+fy′As′+∑NsiRk=α1fcbxh0−x/2+fy′As′h0−as′+∑Msi/e0.0911BendingRk=α1fcbxh0−x/2+fy′As′h0−as′0.0548ShearingRk=Vcs+Vp+0.8fyAsbsinαs+0.8fpyApbsinαpVcs=αcvftbh0+fyvAsv/sh0Vp=0.05Np00.0551The method proposed by ISO 2394 uses the analysis model in design, while the method proposed by EN 1990 uses the probabilistic model of resistance, which is one of the reasons for the difference in their derived results.Figure5 presents the differences between the ISO 2394 method and EN 1990 method for the derivation of values with different structure members with unknown σR. From Figure 5, it can be seen that(1)
It shows that the derived results using the EN 1990:2002 method are lower than those using the ISO 2394:2015 method with unknownσR. The method of EN 1990:2002 is too conservative because the derived results are low and even decrease when the number is low, while ISO 2394 derived high.(2)
The higher the variability of the resistance influencing factors, the lower the derived value of the design resistance, as shown in Figure5(b).(3)
In most cases, the ISO method can derive higher resistance values, but it can only be applied to structural elements with known resistance analysis models.Figure 5
Deriving results when the coefficient of variation ofθ is unknown. (a) Axial compression (short column). (b) Small eccentric compression (short column). (c) Large eccentric compression (short column). (d) Bending. (e) Shearing. Gray denotes the standard curve, blue represents the derivation curve by ISO 2394, and green represents the derivation curve by EN 1990.
(a)(b)(c)(d)(e)The ISO 2394 method is built on the relationship between the design value of resistance and the vector of random variables to obtainRd, which is formula (31). However, since pd in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), Rd derived from formula (31) has a different guarantee rate. To clarify, for different xd, w, ηd, and θd, the guarantee rate of the design value of resistance pd is generally different or even has a large difference, which is difficult to ensure effective control of structural reliability. Therefore, this method of ISO is approximate.The method mentioned in EN 1990 is empirical in character because the derivation ofRk and Rd is artificially weighted and averaged. Besides, the method has the following shortcomings in the derivation process considered.(1)
Formulas (37) and (38) are both established using the Bayesian procedures without setting the confidence level, which is not proper for the derivation of Rd.(2)
Formula (38) incorrectly adopts k∞=z1−pk, but in fact k∞=z1−pk1+1/n. When n is large, it can be considered that k∞=z1−pk, which does not apply to other cases.(3)
Formula (35) does not hold between my and mR. my=μlnR and mR=μR, and μlnR has the following relationship with μR, which does not apply to the small sample condition:(55)μlnR=lnμR1+VR2.(4)
Since formulas (3738) correspond to the condition that VR is completely unknown and completely known, respectively, Q in the two formulas should have been completely unknown and completely known, respectively. However, the mean value of Q in the method mentioned by EN 1900 is established according to Vη in formula (42), which does not fit the case that VR is known and unknown. In summary, although the approximation in the method is very necessary, its theory is not rigorous.


## 4. Conclusions

(1)
The design assisted by the testing is applicable to both partial factor method and design value method, which is the most empirical among the three methods and is one of the important directions in the development of the current structural design methods. However, the design assisted by the testing proposed by current international standards applies solely to the ultimate limit state, so it is necessary to establish the method applicable to serviceability limit state.(2)
The derivation methods of resistance design values are grouped into two categories: the first category of derivation methods is based on resistance variables, including interval estimation, Bayesian procedures, and evaluation using probabilistic methods, of which evaluation using probabilistic methods essentially belongs to Bayesian procedures; the second category of derivation methods based on analytical models, which can reduce the uncertainty in derivation and theoretically lead to better derivation results, is the key derivation method in the derivation of resistance design values.(3)
Both interval estimation method and Bayesian procedures can be used to derive the characteristics and design values of resistance, but the differences in results lead to greater doubts in the choice of methods by designers. Therefore, current international standards only retain the Bayesian method, which is different from the general Bayesian method because of its indirect connection to the basis of derivation. Therefore, a reasonable Bayesian derivation method by comparison with interval estimation method should be studied and established.(4)
The derivation method based on the analytical model in ISO is an indirect computational inferential method based on the relationship between the resistance and each influencing factor, which is very shallow in the probability sense and does not directly reflect the influence of the probability characteristics of each factor. The method proposed by EN belongs to the empirical method utilizing the resistance model only in determining the probabilistic properties. The method is actually built on an extrapolation method based on resistance variables such as formulas (37) and (38), rather than directly on the basis of a probabilistic model of resistance. Both methods are important and have drawbacks in the derivation process, so a rational method based directly on the probabilistic model of resistance should be studied and established.



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