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

Yukun, Hu; Jitao, Yao; Pengfei, Ma; Zhengjie, Cheng; Luyang, Zhang

doi:https://doi.org/10.1155/2022/2198299

Mathematical Problems in Engineering

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Abstract Introduction Analysis Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments Supplementary Materials References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2198299 | https://doi.org/10.1155/2022/2198299

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

Hu Yukun,^1,2Yao Jitao ,^1,2Ma Pengfei,^1,2Cheng Zhengjie,^1,2and Zhang Luyang^1,2

Academic Editor: Antonio Forcina

Received01 Jun 2022

Accepted07 Sept 2022

Published07 Oct 2022

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 the values, the number of tests , the coefficient of variation , and the coefficient of variation of resistance 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.

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 Figure 1, 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].

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 /unknown coefficient [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 Figure 2, 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.

Indirect assessment can be defined as [15]where = the characteristic value of the resistance - implied model uncertainty factor ; = material partial factors; and = the design value of the conversion factor 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 distribution , the characteristic value of the resistance can be expressed as [17]where and = the cumulative probability/probability of non-exceedance of characteristic value of the resistance .

Suppose are resistance samples obtained from bearing capacity test. When the standard deviation of resistance is known, the pivotal quantity obeys the non-central -distribution with df (degree of freedom) and non-centrality parameter , where is the sample mean and is sample standard deviation [30]. According to interval estimation method [31], is defined as the characteristic value of the resistance, which can be obtained aswhere = values of resistance sample; = sample size; = value of sample mean; = value of sample standard deviation; = confidence level; and = the quantile of non-central -distribution with df and non-centrality parameter .

When the standard deviation of resistance is known, the pivotal quantity obeys standard normal distribution . In the same way, the characteristic value is defined aswhere = the quantile of standard normal distribution.

To calculate the characteristic value of the resistance, formula (5) is employed for known and formula (6) is employed for unknown [16].where ; .

Apparently, is improperly replaced with in formula (6). It is proper, as commended by ISO 2394:1998, to give the value of the confidence level less than 0.75 and the value for 0.95 without available information.

It is worth noting that the value of and recommended by ISO 2394:1998 does not apply for structure in China. A value of 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 only by the direct assessment and presume that obeys the normal distribution .

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]:where = be proportional to.

The likelihood function for can be obtained by

The joint posterior distribution of and can be derived by formula (9) using Bayesian formula.

The probability density function for can be obtained by the following formula.where the pivotal quantity obeys Student’s t distribution with df [30]. Then, can be obtained, based on probability definition, bywhere = quantile value of t distribution with df and = the guarantee rate of .

With known, the Jeffreys prior for the unknown parameter is as follows [32, 33]:

Similar to the Bayesian derivation, the pivotal quantity obeys the standard normal distribution. was performed in the same way as described above:

Since the value, derived by the formulas (11) and (13), is under laboratory conditions, derivation of proposed by ISO 2394 derived by formulas (14) and (15) includes the conversion factor 16, 17:

The above formula is direct derivation pathway, while the indirect derivation pathway is as follows:

Similarly, is improperly replaced with in formulas (14–16). Furthermore, the coefficient is only dependent on the sample size, , and on the chosen guarantee rate, , , without the confidence level .

(2) Eurocode: Basis of Structural Design. The method proposed by EN is similar to ISO 2394, and derivation of is as follows [15]:where = the coefficient of variation of ; if is known, ; if is unknown, ; , depend on the sample size, , and on the chosen guarantee rate, .

This method of deriving the is similar to the approach proposed by ISO 2394 [16, 17]. However, the row “ known” should represent “ known” but should actually represent “,” whose derived results are completely different.

As mentioned in EN 1990:2002, the resistance obeys the log-normal distribution . So, obeys the normal distribution and the characteristic parameter estimated according to formulas (19 20). With the log-normal distribution, the direct and indirect assessment of becomes formulas (21 22).

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 of may even appear to be negative with small sample size and large guarantee rate and variability. Fortunately, to avoid this fact, the resistance 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 parameters and 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 that , which is the conditional probability distribution about , obeys the normal distribution , and obeys the inverse gamma distribution , and the probability density function of is

Also, the mean value and coefficient of variation of are determined by formulas (25) and (26). When the value of is large, the mean value and coefficient of variation of are approximated according to equations (27) and (28), respectively.

If it is assumed that a series of a priori tests have been conducted before the test, and are equivalent to the point estimates of and , respectively, is equivalent to the sample size in deriving , and is equivalent to the degree of freedom in deriving . 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:where = the quantile of -distribution with df and = guaranteeing rate of the characteristic or design values. When the standard deviation of the resistance is known, the prior distribution of is taken to be the normal distribution and the resistance is determined according to the following formula:

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]:where = the design value of vector of random variables; = the value of measurable deterministic variables; = the design value of the conversion coefficient - misnamed the design value of coefficient of the model uncertainty; ; and = 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 is considered to be a random variable assuming a log-normal distribution and its design value is estimated fromwhere ; = mean deviation for ; = standard deviation for ; and = 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 obtain , which is formula (31). However, Since p_d in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), R_d derived from formula (31) has a different guarantee rate. To clarify, for different , , , and , the guarantee rate of the design value of resistance 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 derive but also derive . Since the vector of random variable is difficult to determine, the method cannot derive and from the ideal row of known and unknown. Therefore, to solve the above problem, the method introduces a weighted average.

As obeys the log-normal and or is unknown, the Bayesian procedure mentioned by EN 1990 derives according to the following formula:where = mean value of the logarithm of the measured resistance values; = standard deviation of the logarithm of the measured resistance values; and ( unknown), with no information ().

Set

Hence [2]:

As or is unknown, the Bayesian procedure mentioned by EN 1990 derives according to the following formula [16]:where when n tends to infinity, [, then ].

The probabilistic mode of the resistance was constructed as follows [16]:where = probabilistic structural performance function implied correction factor ; = the model uncertainty; and = theoretical value of resistance.

Suppose and both obey log-normal distribution with the following formulation:where = the coefficient of variation of and = the coefficient of variation of .

Because generally is known and is unknown, the derivation of lies between the calculated values of formula (37) (V_R known) and formula (38) ( unknown). Consequently, the weighting factors can be defined as follows [16]:

Obviously, the larger , the higher the unknown degree of . The larger , the higher the known degree of . Following this weighted average method, the derivation formula for is as follows:where = values of model uncertainty sample. If , . Then, the derivation of be obtained by the following formula:

By the same weighted average method, the derivation formula for the value that implicitly accounts for is obtained as [16]

When is greater than or equal to 100, the design value of resistance was calculated according to the formula as follows.where ; ; and .

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 deriving by indirect assessment with 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, , defined by formulas (49) and (50).

Table 1 presents details on the statistical parameters of resistance of structural members, where is calculated from formula (47), and other parameters are obtained from GB 50153-2008 [18].

Figure 3 presents the differences between the interval estimation method and Bayesian procedures for the derivation of values with different structure members with unknown . Figure 4 presents the differences with known . 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 tests is small and value is large.(2)Compared to interval estimation, Bayesian procedures tend to be conservative with smaller sample size and larger guarantee rate and adventurous in other cases.(3)With a small number of tests , 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 variation , the derived results are better than unknown coefficients of variation, especially for a small number of tests .

(a)

(b)

(c)

(d)

(e)

(f)

(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 sample 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 Section 3.1, 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 is calculated by formula (53). If the resistance function is a more complex functional form, then 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 values [18].

The 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.

Figure 5 presents the differences between the ISO 2394 method and EN 1990 method for the derivation of values with different structure members with unknown . 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 unknown . 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 Figure 5(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.

(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 obtain , which is formula (31). However, since in the current partial factor method is uncertain and varies widely (due to the lack of sufficient reliability control), derived from formula (31) has a different guarantee rate. To clarify, for different , , , and , the guarantee rate of the design value of resistance 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 of and 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 .(2)Formula (38) incorrectly adopts , but in fact . When is large, it can be considered that , which does not apply to other cases.(3)Formula (35) does not hold between and . and , and has the following relationship with , which does not apply to the small sample condition:(4)Since formulas (37 38) correspond to the condition that is completely unknown and completely known, respectively, in the two formulas should have been completely unknown and completely known, respectively. However, the mean value of in the method mentioned by EN 1900 is established according to in formula (42), which does not fit the case that 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.

Data Availability

The structural component statistics data used to support the findings of this study are included within the supplementary information file.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Y.K.H. and J.T.Y. were responsible for conceptualization, methodology, and formal analysis. J.T.Y. was responsible for validation, resources, supervision, and funding acquisition. Y.K.H. was responsible for data curation and original draft preparation. P.F.M. and Y.K.H. were responsible for software. P.F.M. was responsible for investigation. Y.K.H. and L.Y.Z. were responsible for review and editing. Z.J.C. was responsible for visualization. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This research was funded by the Natural Science Foundation of China (grant no. 51278401).

Supplementary Materials

In order to acquire the values of model uncertainty sample for five types of structural members, a database of 68 axial compression members, 32 eccentricity loaded members with tension failure, 40 eccentricity loaded members with compression failure, 44 bending members, and 42 shear members is collected and listed in Supplementary Material A. (Supplementary Materials)

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