Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength

Jiao Wang; Weiji Zheng; Yangang Zhao; Xiaogang Zhang

doi:10.3390/app10030815

Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength

Wang, Jiao;Zheng, Weiji;Zhao, Yangang;Zhang, Xiaogang 2020-01-23 00:00:00 applied sciences Article Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength 1 1 , 2 3 Jiao Wang , Weiji Zheng * , Yangang Zhao and Xiaogang Zhang School of Civil Engineering, Guangzhou University, Guangzhou 510006, China; cewangjiao@e.gzhu.edu.cn Department of Architecture and Building Engineering, Kanagawa University, Yokohama 2218686, Japan; zhao@kanagawa-u.ac.jp College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China; szzxg@szu.edu.cn * Correspondence: 2111816256@e.gzhu.edu.cn Received: 15 December 2019; Accepted: 21 January 2020; Published: 23 January 2020 Abstract: Since the heterogeneity of the cement-based material contributes to a random spatial distribution of compressive strength, a reliability analysis based on the compressive strength of concrete is fundamental to carry out structural safety assessment. By analyzing 10,317 datapoints on compressive strength of concrete, a time-varying reliability evaluation based on the third-moment (TM) method was proposed to predict the service life of concrete. Unlike the second-moment (SM) method, skewness is taken into account in the TM; thus, the calculated result of concrete failure time based on the TM is more accurate. In this paper, the errors of the calculated results using time-varying reliability evaluation are within 3%, as shown by Monte Carlo (MC) simulation. In addition, the proposed model (aiming to calculate the equivalent design compressive strength) veriﬁes the concrete failure time calculated by time-varying reliability. According to the results, concrete failure times calculated by these two models are in good agreement. Overall, based on the simple and eective methodology adopted in this paper, it is feasible to develop time-varying reliability based on other factors that might also lead to concrete failure, such as carbonation-induced corrosion, cracking, or deﬂection of the concrete. Keywords: concrete strength; failure probability; third-moment method; equivalent design concrete strength 1. Introduction Reinforced concretes have been applied worldwide in structural engineering since the beginning of the nineteenth century [1–5]. However, structure damage has increasingly occurred under aggressive service environments, which might be caused by ambient temperature, freeze–thaw cycles, precipitation, carbon dioxide, chloride, and sulfate [6–8]. Therefore, it is signiﬁcant to implement eective strategies to guarantee structural safety and strengthen vulnerable buildings. In this paper, a proposed model based on the compressive strength of concrete was applied to predict the concrete failure time. Carbonation is one of the most fundamental triggers of concrete deterioration [9–13], and carbonation depth is used to describe the level of carbonation-induced corrosion. As presented in the existing paper, the carbonation depth has both favorable and unfavorable eects on the compressive strength of concrete. On the positive side, carbonation-induced corrosion contributes to the increase of the compressive strength and the harnessing of concrete because numerous concrete pores are occupied by the carbonation product, calcium carbonate [14]. On the negative side, too much calcium carbonate that exceeds the limit of concrete pores might cause additional internal pressure and Appl. Sci. 2020, 10, 815; doi:10.3390/app10030815 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 815 2 of 18 micro-cracking [15,16]. In this paper, the relationship between carbonation depth and the compressive strength of concrete is analyzed using a prediction model of carbonation depth [17]. Since the presence of variations associated with climatic factors (e.g., temperature and humidity), concrete materials, and geometric conﬁgurations [18,19], the compressive strength of concrete should be regarded as a variable in order to account for these uncertainties in deterioration. In practical engineering problems, these unavoidable uncertainties are often qualiﬁed by probabilistic models [20–22], among which reliability analyses are widely applied in structural safety assessments [23–25]. In the existing time-varying reliability models, the probabilities of corrosion initiation and corrosion damage are calculated based on the change in climatic variables [26]. Szilágyi [27] developed a phenomenological constitutive model for obtaining the rebound hardness of a surface concrete as a property of the time-dependent material. Nevertheless, this model has not yet been ratiﬁed for practical applications. A previously conducted study [28] exhibits the property of compressive strength in the case of deteriorating concrete in the Tohoku region of Japan; however, the prediction model of compressive strength and the method of structural safety evaluation were not provided. Generally, most existing probabilistic models may not be suitable for evaluating the service life of concrete. Based on a large number of tested data on compressive strength, an eective model based on the reliability to evaluate the service life of concrete was developed. It is noted that dierent sizes and shapes of the tested components aect the compressive strength of concrete. Therefore, the conversion and standardization of the value of compressive strength may produce an inaccurate result [29]. In this paper, all the shapes and sizes of the experimental members were uniﬁed. Additionally, the compressive strengths of all the samples were tested using core pulling. Gao [30] proposed that the compressive strength obeys a normal distribution; however, by analyzing 10,317 datapoints considered in this study, it is observed that a gamma distribution is more appropriate to describe the distribution of compressive strength. Moreover, though Gao’s model serves the purpose of evaluating the compressive strength, only the mean value of compressive strength is considered. Therefore, the probabilistic model may not obtain the precise prediction of concrete failure time. Regarding the discreteness of the statistic, both mean value and coecient of variation are expected to be considered in the model to evaluate the service life of concrete. Thus, a time-varying evaluation model based on reliability was proposed in this study. The failure probability and reliability index equivalently evaluate the structural safety. However, the reliability index, being very simple and with respect to function, is not integral, which is in contrast to the calculation of failure probability. Among all the methods for computing the reliability index, moment methods are used worldwide to observe the reliability index [25]. Compared with the second-moment (SM) method, the reliability evaluation based on the third-moment (TM) method is more accurate because the skewness (the third dimensionless central moment) is considered [31,32]. Tested by MC simulation, the results evaluated by the TM reliability are more accurate than those by the SM reliability. In this paper, the concrete guaranteed rate was applied to predict the concrete failure time calculated by time-varying reliability. To testify the reliability-based computation of the concrete failure time, the equivalent design compressive strength (functioning as the ultimate concrete compressive strength) was investigated to observe the concrete failure time. The concrete failure times computed by both models agree well. 2. Distribution Fitting of the Compressive Strength of Concrete Deterioration in concrete construction is a severe problem that leads to durability problems and even structural failure. In order to prevent concrete failure and retroﬁt the aging constructions over time, periodical testings of the structural components are conducted every three years, as shown in Figure 1. In this paper, all data were obtained from these testings, and all tested components were of natural ﬁne aggregated concrete, taken from 1061 buildings (e.g., schools, residences, and oce buildings) in Yokohama, Japan, where the average temperature and humidity were 15.208 C and Appl. Sci. 2020, 10, 815 3 of 20 Appl. Sci. 2020, 10, 815 3 of 18 Appl. Sci. 2020, 10, 815 3 of 20 core compressive strength ( ), and the average dimensions of the components were 80 mm in diameter Appl. and Sci. 10 2020 0 mm in hei , 10, 815 ght. 3 of 20 core compressive strength ( ), and the average dimensions of the components were 80 mm in diameter 69%, respectively. The tested compressive strength in these experiments was regarded as the core and 100 mm in height. compressive strength ( f ), and the average dimensions of the components were 80 mm in diameter cor core e compressive strength ( ), and the average dimensions of the components were 80 mm in diameter and 100 mm in height. and 100 mm in height. Figure 1. The compressive strength experiment of concrete core. The tested data (10,317) were divided into several groups according to different design values Figure 1. The compressive strength experiment of concrete core. of concrete strength ( Figure 1. Figure ) and t 1. The c The he compr o service mpressive essive tim str str e of e ength ng concr th ex experiment periment of ete (yearof s),concrete concr as sh ete own cor core. in Fig e. ures 2 and 3. The tested data (10,317) were divided into several groups according to different design values The tested data (10,317) c were divided into several groups according to dierent design values of The tested data (10,317) were divided into several groups according to different design values of concrete strength ( ) and the service time of concrete (years), as shown in Figures 2 and 3. concrete strength ( f ) and the service time of concrete (years), as shown in Figures 2 and 3. c f of concrete strength ( ) and the service time of concrete (years), as shown in Figures 2 and 3. f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). Figure 2. Statistics of concrete strength ( f = 18.0N/mm ). c 2 f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). Figure 3. Statistics of concrete strength ( f = 21.0N/mm ). Appl. Sci. 2020, 10, 815 4 of 20 Appl. Sci. 2020, 10, 815 4 of 20 f = 21.0 N/mm c 2 Figure 3. Statistics of concrete strength ( ). f = 21.0 N/mm Appl. Sci. 2020, 10, 815 c 4 of 18 Figure 3. Statistics of concrete strength ( ). MATLAB software was used to analyze the fitting distributions of datasets with the sample MATLAB software was used to analyze the fitting distributions of datasets with the sample significanc MATLAB e level a = softwar 0.05. e was The used tests oto f ganalyze amma distri the b ﬁtting ution, n distributions ormal distribution of datasets , Rayle with igh distribution, the sample significance level a = 0.05. The tests of gamma distribution, normal distribution, Rayleigh distribution, signiﬁcance and extreme level value d a = 0.05. istribThe ution wer tests of e e gamma xecuted b distribution, ased on these da normal distribution, ta. It was foRaylei und tha ght the distribution, gamma and extreme value distribution were executed based on these data. It was found that the gamma and distri extr buti eme on ha value d the hi distribution ghest goodness of were executed fit, as sh based own in onFigure 4. Figure 5 these data. It was shows the found that comparison the gamma of distribution had the highest goodness of fit, as shown in Figure 4. Figure 5 shows the comparison of distribution R-squares bet had ween the gamma distr the highest goodness ibution and the of ﬁt, as shown noin rmal d Figuristr e 4i.bution of d Figure 5 shows ifferent service the comparison years. R-squares between the gamma distribution and the normal distribution of different service years. of Accordi R-squar ng to Fi es between gure 5, the ga the gamma mma distribution distributi and on proved to be more a the normal distributionppl ofidi cabl ereent to describe the service years. According to Figure 5, the gamma distribution proved to be more applicable to describe the Accor compressi dingve to Figur strength. e 5, the Addi gamma tionadistribution lly, for a deteri proved orating proce to be moress, applicable the stoch to astic describe gamma proce the comprss w essive as compressive strength. Additionally, for a deteriorating process, the stochastic gamma process was adopted to analyze the time-varying reliability. Abdel-Hameed [33] was the first to propose the strength. Additionally, for a deteriorating process, the stochastic gamma process was adopted to adopted to analyze the time-varying reliability. Abdel-Hameed [33] was the first to propose the analyze gamma process the time-varying as the first reliability choice to descri . Abdel-Hameed be the pe[rfor 33] ma was ncthe e dﬁrst egrad to atpr ioopose n procthe ess [34]. Mo gamma pr reover, ocess gamma process as the first choice to describe the performance degradation process [34]. Moreover, van Noortwijk modeled the stochastic process of deterioration with a gamma process, which had as the ﬁrst choice to describe the performance degradation process [34]. Moreover, van Noortwijk van Noortwijk modeled the stochastic process of deterioration with a gamma process, which had modeled independent the gamma-d stochastic istributed inc process of deterioration rements with ide with ntia cagamma l scale pa pr raocess, meters [35 which ]. The resul had independent ts indicate independent gamma-distributed increments with identical scale parameters [35]. The results indicate that a gamma process can properly model the monotonic behavior of aging. Overall, the preceding gamma-distributed increments with identical scale parameters [35]. The results indicate that a gamma that a gamma process can properly model the monotonic behavior of aging. Overall, the preceding pr rese ocess arch can figpr ure shows operly model that t the he gamma monotonic process behav is ior s ofuit aging. able tOverall, o model the gra pr du eceding al and monot research onica ﬁgur lly e research figure shows that the gamma process is suitable to model gradual and monotonically accumulating damage over time [35,36]. Based on these existing conclusions and the analysis in this shows that the gamma process is suitable to model gradual and monotonically accumulating damage accumulating damage over time [35,36]. Based on these existing conclusions and the analysis in this over paper, the time [ gamma 35,36]. distribution is a Based on theseppropria existing tconclusions e to describe the d and the istribution of co analysis in this mpressive paper, the stregamma ngth. paper, the gamma distribution is appropriate to describe the distribution of compressive strength. distribution is appropriate to describe the distribution of compressive strength. Figure 4. Distribution fitting of compressive strength. Figure 4. Distribution ﬁtting of compressive strength. Figure 4. Distribution fitting of compressive strength. Gamma Gamma 1.02 1.02 Normal Normal 0.98 0.98 0.96 0.96 0.94 0.94 0.92 0.92 0.9 0.9 0.88 0.88 Time (year) 35 36 37 38 39 40 41 42 43 44 Time (year) 35 36 37 38 39 40 41 42 43 44 Figure 5. Contrast of goodness of ﬁt for gamma distribution and normal distribution. Figure 5. Contrast of goodness of fit for gamma distribution and normal distribution. Figure 5. Contrast of goodness of fit for gamma distribution and normal distribution. 3. Prediction for the Mean Value of Compressive Strength 3. Prediction for the Mean Value of Compressive Strength R-square R-square Appl. Sci. 2020, 10, 815 5 of 18 3. Prediction for the Mean Value of Compressive Strength Carbonation-induced corrosion can lead to a residual capacity loss of the reinforced concrete structures [37]. Generally, carbonation depth is used to deﬁne the carbonized level of the concrete. To analyze the relation between the carbonation depth and the compressive strength of concrete, and to develop a prediction model for the mean value of compressive strength (MVCS) in which the environment factors (temperature and humidity) are considered, the ratio of compressive strength to carbonation depth was investigated. In addition, the concrete failure time calculated by this prediction model was used to compare with those calculated by time-varying reliability. Based on the existing model of carbonation depth [17], the carbonation depth is expressed as 0.05 f cm D(t) = 7.742e t (1) T RH 0.25 = (T/T ) (2) T 0 = RH(100 RH)/RH (100 RH ) (3) RH 0 0 f = f /(1 1.645) (4) cm cu,k where D is the mean value of carbonation depth in a nonstandard condition (mm); is the coecient of temperature; is the coecient of humidity; T is the environmental temperature ( C); T is the RH 0 average temperature ( C); RH is the environmental humidity (%); RH is the average humidity (%); t is the service time of concrete (year); f is the mean value of the standard value of compressive strength cm 2 2 (N/mm ); f (equal to concrete grade) is the standard value of compressive strength (N/mm ), cu,k which is observed by converting f (design value of compressive strength (N/mm )) [38]; and is the coecient of variation, which can be obtained via the Code for Design of Concrete Structures [39]. According to the investigation of the relevant climate data, the climate changes among dierent years were not apparent. In order to simplify the calculation of Equation (1), T is set as the median of all mean values of the temperature of dierent years, and RH is the median of all mean values of the humidity of dierent years. Dierent from the prediction model of carbonation depth in the existing paper [17], f is replaced by f , which is more appropriate to calculate the ratio Ra(t), as shown in cu,k cm Equation (4). Relevant parameters calculating the carbonation depth are presented in Table 1. Table 1. Relevant parameters calculating the carbonation depth in Yokohama, Japan. 2 2 Relevant Parameter f =18.0 N/mm f =21.0 N/mm c c T( C) 15.425 15.425 T ( C) 15.208 15.208 RH(%) 62 62 RH (%) 69 69 f (N/mm ) 37.6 45 cu,k 0.16 0.15 f (N/mm ) 51 59.8 cm To analyze the relation between compressive strength of concrete and carbonation depth, the ratio of relative concrete strength to carbonation depth is developed as Ra(t) = Rs(t)/D(t) (5) Rs(t) = (t)/ f (6) Rs c where Ra is the ratio of the relative MVCS to the mean value of carbonation depth (MVCD); Rs is the relative MVCS; and is the MVCS of dierent years (N/mm ), which is obtained based on 10,317 Rs experimental datapoints, as shown in Table 2. Appl. Sci. 2020, 10, 815 6 of 20 Rs(t ) = μ (t ) / f (6) Rs c Ra Rs where is the ratio of the relative MVCS to the mean value of carbonation depth (MVCD); is Rs N / mm the relative MVCS; and is the MVCS of different years ( ), which is obtained based on 10,317 experimental datapoints, as shown in Table 2. 2 2 f = 21. 0 N/mm f = 18. 0 N/mm c c Table 2. Mean value of compressive strength with (a) and (b) . Appl. Sci. 2020, 10, 815 6 of 18 (a) Mean Value of Compressive Strength with f = 21.0 N/mm 2 2 Time (Year) 35 36 37 38 39 40 41 42 43 44 45 Table 2. Mean value of compressive strength with (a) f = 21.0 N/mm and (b) f = 18.0 N/mm . c c Rs1 31.95 3 (a)0 Mean .70 V3 alue0.19 of Compressive 31.29 28Strength .04 25with .27 2 f =3.78 21.0 N/2 mm3.71 23.54 24.51 27.43 N/ mm ( ) Time (Year) 35 36 37 38 39 40 41 42 43 44 45 (b) Mean Value of Compressive Strength with f = 18.0 N/mm 2 c (N/mm ) 31.95 30.70 30.19 31.29 28.04 25.27 23.78 23.71 23.54 24.51 27.43 Rs1 Time (Year) 42 43 44 45 46 47 48 49 50 51 52 (b) Mean Value of Compressive Strength with f = 18.0 N/mm Rs Time (Year) 42 43 44 45 46 47 48 49 50 51 52 22.65 23.00 24.15 23.34 23.14 22.96 22.88 21.54 22.47 22.19 19.73 N / mm 2 ( ) 22.65 23.00 24.15 23.34 23.14 22.96 22.88 21.54 22.47 22.19 19.73 (N/mm ) Rs2 Time (year) 53 54 Time (year) 53 54 μ 2 (N/mm ) 20.77 20.03 Rs2 Rs2 20.77 20.03 N / mm ( ) According to Table 2, it was found that the compressive strength of concrete decreased with the According to Table 2, it was found that the compressive strength of concrete decreased with the increase of time. increase of time. Based on Equations (1)–(6) and Tables 1 and 2, Ra values of dierent servicing years were ﬁtted as Based on Equations (1)–(6) and Tables 1 and 2, Ra values of different servicing years were fitted shown in Figure 6. as shown in Figure 6. Figure 6. Fitting of the ratio of the relative concrete strength to the mean value of carbonation depth. Figure 6. Fitting of the ratio of the relative concrete strength to the mean value of carbonation depth. According to Figure 6, Ra decreased as time increased, showing that the compressive strength According to Figure 6, Ra decreased as time increased, showing that the compressive strength of concrete decreased with the increase of carbonation depth over time. In addition, it is known that of concrete decreased with the increase of carbonation depth over time. In addition, it is known that the carbonation depth increased with the increase of time. Therefore, combined with the deﬁnition of the carbonation depth increased with the increase of time. Therefore, combined with the definition carbonation depth, it was concluded that carbonation-induced corrosion can result in the decrease of of carbonation depth, it was concluded that carbonation-induced corrosion can result in the decrease compressive strength of concrete, which agrees with the conclusion drawn from Zha [40]. of compressive strength of concrete, which agrees with the conclusion drawn from Zha [40]. The ﬁtting curves in Figure 6 are expressed as The fitting curves in Figure 6 are expressed as Ra (t) = 0.02t + 1.279 (7) Ra (t) = 0.0058t + 0.5445 (8) where Ra is the ratio of the relative compressive strength to carbonation depth with f = 21.0 N/mm ; and Ra is the ratio of the relative compressive strength to carbonation depth with f = 18.0 N/mm . 2 c By comparing the changing rate of Ra (t) with Ra (t), the deterioration rate of concrete with 2 2 f = 21.0 N/mm is faster than that of f = 18.0 N/mm . c c Appl. Sci. 2020, 10, 815 7 of 20 Ra (t ) = −0.02t + 1.279 (7) Ra (t ) = −0.0058t + 0.5445 (8) Ra where is the ratio of the relative compressive strength to carbonation depth with Ra ; and is the ratio of the relative compressive strength to carbonation depth with f = 21.0 N/mm f = 18.0 N/mm . Appl. Sci. 2020, 10, 815 7 of 18 Ra (t ) Ra (t) 1 2 By comparing the changing rate of with , the deterioration rate of concrete 2 2 with f = 21.0 N/mm is faster than that of f = 18.0 N/mm . Onc the basis of Equations (1), (5), (7) and c (8), the new prediction models of Rs are proposed as Rs On the basis of Equations (1), (5), (7) and (8), the new prediction models of are proposed as 0.05 f cm1 Rs (t) = D (t) Ra (t) = 7.742e t (0.02t + 1.279) (9) 1 1 1 T RH __ − 0.05 f cm 1 (9) Rs (t ) = D (t ) × Ra (t ) = 7.742 e × t × α × α × (−0.02t + 1.279 ) 1 1 T RH 0.05 f cm2 Rs (t) = D (t) Ra (t) = 7.742e t (0.0058t + 0.5445) (10) 2 2 2 T RH __ − 0.05 f where Rs is the relative compressive strength of cm 2 f = 21.0 N/mm ; Rs is the relative compressive (10) 1 c 2 Rs (t ) = D (t ) × Ra (t ) = 7.742e × t × α × α × (−0.0058t + 0.5445) 2 2 T RH strength of f = 18.0 N/mm ; f is the mean value of the standard value of compressive strength cm1 Rs Rs with f = 21.0 N/mm ; and f is the mean value of the standard value of compressive strength with 1 2 where c is the relative com cmp 2 ressive strength of ; is the relative compressive f = 21.0 N/mm f = 18.0 N/mm . 2 cm 1 strength of ; is the mean value of the standard value of compressive strength f = 18.0 N/mm Figure 7 is c developed from Equations (9) and (10), showing that with an increase of service time, the compressive strength decreased, and the speed of strength deterioration increased. Rs(t) = 1.0 is cm 2 2 with ; and is the mean value of the standard value of compressive strength with f = 21.0 N/mm the limit state of concrete failure in conducting concrete safety assessments (failure mode I). As shown f = 18.0 N/mm . c 2 in Figure 7, the concrete failure time of f = 21.0 N/mm was observed to be 48.2 years, while that Figure 7 is developed from Equations (9) and (10), showing that with an increase of service time, of f = 18.0 N/mm was observed to be 61.8 years. However, the coecient of variation was not the compressive strength decreased, and the speed of strength deterioration increased. considered in this model, leading to the inaccurate prediction of concrete failure time. Rs (t ) = 1 . 0 Figures 8–10 also present the corresponding ﬁtting curves with acceptable R-squares, indicating is the limit state of concrete failure in conducting concrete safety assessments (failure that the inﬂuence of f on the coecient of variation, the skewness, and the kurtosis was negligible. mode Ι). As shown in Figure 7, the concrete failure time of was observed to be 48.2 f = 21.0 N/mm Based on Figures 8–10, three eigenvalues increased with the increase of time. Besides, the coecient of years, while that of f = 18.0 N/mm was observed to be 61.8 years. However, the coefficient of variation increased over time, resulting in dispersion expansion of the statistic. Therefore, the accuracy variation was not considered in this model, leading to the inaccurate prediction of concrete failure of the observed concrete failure time determined only by the mean value would decrease when time. time increases. 2 2 Figure 7. Relative concrete strength ( f = 18.0N/ mm and f = 21.0 N/mm ). c c 2 2 Figure 7. Relative concrete strength ( f = 18.0 N/mm and f = 21.0 N/mm ). c c 4. Eect of f on the Distribution of Compressive Strength Based on the analysis of distribution ﬁtting, a gamma distribution was used to describe the 4. Effect of on the Distribution of Compressive Strength distribution of compressive strength. To investigate the eect of f on the distribution, the coecient of variation, skewness, and kurtosis of tested data with dierent f values were investigated, as presented in Figures 8–10. Appl. Sci. 2020, 10, 815 8 of 20 Appl. Sci. 2020, 10, 815 8 of 20 Based on the analysis of distribution fitting, a gamma distribution was used to describe the Based on the analysis of distribution fitting, a gamma distribution was used to describe the distribution of compressive strength. To investigate the effect of on the distribution, the distribution of compressive strength. To investigate the effect of on the distribution, the coefficient of variation, skewness, and kurtosis of tested data with different values were investigated, as presented in Figures 8–10. coefficient of variation, skewness, and kurtosis of tested data with different values were Appl. Sci. 2020, 10, 815 8 of 18 investigated, as presented in Figures 8–10. Figure 8. Time-varying coefficient of variation. Figure Figure 8. 8. T Time- ime-varying varying coefficient of coecient of variation. variation. Appl. Sci. 2020, 10, 815 9 of 20 Figure 9. Time-varying skewness. Figure 9. Time-varying skewness. Figure 9. Time-varying skewness. Figure 10. Time-varying kurtosis. Figure 10. Time-varying kurtosis. Figures 8–10 also present the corresponding fitting curves with acceptable R-squares, indicating that the influence of on the coefficient of variation, the skewness, and the kurtosis was negligible. Based on Figures 8–10, three eigenvalues increased with the increase of time. Besides, the coefficient of variation increased over time, resulting in dispersion expansion of the statistic. Therefore, the accuracy of the observed concrete failure time determined only by the mean value would decrease when time increases. According to the property of gamma distribution, these three eigenvalues can be calculated by the shape parameter , which are expressed as α λ 1 V = × = (11) λ α α = (12) α = (13) α α V 3 4 where is the coefficient of variation; is the shape parameter; is the skewness; and is the kurtosis. Since the three eigenvalues are determined only by , it is concluded that also has no influence on . 5. Scale Parameter and Shape Parameter of Gamma Distribution A reliability analysis was applied in this paper to predict the concrete failure time with high accuracy. In order to develop a time-varying reliability, the parameters ( and ) of the gamma α λ distribution function at different service years need to be investigated. Different from , varies with different values of . The parameters and the corresponding fitting curves are observed in Figures 11 and 12. Appl. Sci. 2020, 10, 815 9 of 18 According to the property of gamma distribution, these three eigenvalues can be calculated by the shape parameter , which are expressed as V = = p (11) = (12) = (13) where V is the coecient of variation; is the shape parameter; is the skewness; and is the kurtosis. Since the three eigenvalues are determined only by , it is concluded that f also has no inﬂuence on . 5. Scale Parameter and Shape Parameter of Gamma Distribution A reliability analysis was applied in this paper to predict the concrete failure time with high accuracy. In order to develop a time-varying reliability, the parameters ( and ) of the gamma Appl. Sci. 2020, 10, 815 10 of 20 distribution function at dierent service years need to be investigated. Dierent from , varies with dierent values of f . The parameters and the corresponding ﬁtting curves are observed in Figures 11 Appl. Sci. 2020, 10, 815 10 of 20 and 12. Figure 11. Scale parameter ( ) fitting ( ). f = 18.0 N/mm Figure 11. Scale parameter ( ) ﬁtting ( f = 18.0 N/mm ). λ 1 Figure 11. Scale parameter ( ) fitting ( ). f = 18.0 N/mm Figure 12. Shape parameter () ﬁtting. Figure 12. Shape parameter ( ) fitting. Figure 12. Shape parameter ( ) fitting. Equations (14) and (15) are the predicted models of these two parameters obtained by curve fitting. Equations (14) and (15) are the predicted models of these two parameters obtained by curve λ (t ) = 12 .94 × exp (−0.072t ) (14) fitting. λ α ( (tt ) ) = = 12 428.4 .94 ××exp exp(− ( − 00 .072 .08tt ) ) (1 (14) 5) Another method for computing the time-varying scale parameter is proposed without fitting α (t ) = 428.4 × exp ( −0.08t ) (15) λ (t ) . The proposed method is presented in the subsequent section. Another method for computing the time-varying scale parameter is proposed without fitting In this paper, MVCS was computed by Equation (16-a). Besides, based on the property of gamma λ (t ) . The proposed method is presented in the subsequent section. distribution, MVCS can also be expressed as Equation (16-b). In this paper, MVCS was computed by Equation (16-a). Besides, based on the property of gamma μ (t) = Rs (t) × f distribution, MVCS can also be expressed as Equation (16-b). (16-a) Rs n c μ (t) = Rs (t) × f (16-a) Rs n c μ (t ) = α (t ) / λ (t ) (16-b) g n μ (t ) = α (t ) / λ (t ) μ μ (16-b) g n Rs where and are both the mean value of compressive strength; and n is used to distinguish Rs λ μ μ c and Rs with different . where and are both the mean value of compressive strength; and n is used to distinguish f μ Rs λ c μ g and with different . Rs Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed based on Equations (10), (16-a) and (14), (15), (16-b), respectively. The difference between μ g Rs Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed based on Equations (10), (16-a) and (14), (15), (16-b), respectively. The difference between Appl. Sci. 2020, 10, 815 10 of 18 Equations (14) and (15) are the predicted models of these two parameters obtained by curve ﬁtting. (t) = 12.94 exp(0.072t) (14) (t) = 428.4 exp(0.08t) (15) Another method for computing the time-varying scale parameter is proposed without ﬁtting (t). The proposed method is presented in the subsequent section. In this paper, MVCS was computed by Equation (16a). Besides, based on the property of gamma distribution, MVCS can also be expressed as Equation (16b). (t) = Rs (t) f (16a) Rs n c (t) = (t)/ (t) (16b) g n where and are both the mean value of compressive strength; and n is used to distinguish Rs and Rs g with dierent f . Appl. Sci. 2020, 10, 815 c 11 of 20 Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed Rs g c based on Equations (10), (16a) and (14), (15), (16b), respectively. The dierence between the predicted Rs the predicted results exists because and were both obtained based on curve fitting. The results exists because and were both obtained based on curve ﬁtting. The errors between the Rs g errors between the predicted results were less than 2%. predicted results were less than 2%. Figure 13. Comparison of results calculated by and ( f = 18.0 N/mm ). g c Rs μ g Rs Figure 13. Comparison of results calculated by and ( f = 18.0 N/mm ). According to Figure 13, it is concluded that and were equivalent to evaluate the compressive Rs g strength, which is expressed as Equation (17a). Therefore, an approximate expression, formulated based on Equation (17a) for the time-varying scale parameter, is elaborated as presented by Equation (17b). Rs According to Figure 13, it is concluded that and were equivalent to evaluate the compressive strength, which is expressed as Equation (17-a). Therefore, an approximate expression, (t) formulated based on Equation (17-a) forRs the t (ti)me-v f ar =ying scale parameter, is elaborated as present (17a) ed n c (t) by Equation (17-b). (t) α (t) (t) = (17b) Rs (t) ×Rs f =(t) f n c (17-a) n c λ (t) In light of Equation (17b), it is feasible to obtain without curve ﬁtting. By using the proposed method, of f = 21.0 N/mm is obtained based on Equations (9), (15), and (17b), as shown in α (t) λ (t) = Figure 14. (17-b) Rs (t) × f n c In light of Equation (17-b), it is feasible to obtain without curve fitting. By using the proposed method, of is obtained based on Equations (9), (15), and (17-b), as shown in f = 21.0 N/mm Figure 14. Appl. Sci. 2020, 10, 815 11 of 18 Appl. Sci. 2020, 10, 815 12 of 20 Figure 14. Scale parameter ( ) with f f = =21.0N 21.0N//mm mm . 2 2 c c Figure 14. Scale parameter ( ) with . 6. Time-Varying Reliability Evaluation Based on the Tm Method 6. Time-Varying Reliability Evaluation Based on the Tm Method Since concrete failure is regarded as f < f , the performance function is deduced as Equation (18), core c f ＜f core c wherSince e x is the concret random e failur variable e is reg of arded compr as essive strength. , the perf Accor orm ding ance tofthe uncti cumul on isative deduced as Eq density function uation (1 (CDF) 8), where of the xgamma is the ra distribution, ndom variable the of failur compressiv e probability e strengt (p ) h is. Ac expr cording t essed aso the cumulative density function (CDF) of the gamma distribution, the failure probability ( ) is expressed as G(x) = x 18 (18) G(x) = x − 18 Z (18) a(t)1 (t)x p (t) = x e dx (19) f f a(t)1 a (t ) −1 − λ (t ) x (t) G((t)) p (t ) = x e dx (19) −a (t ) λ ( t ) Γ (α (t )) where G(x) is the performance function; and x is the random variable of compressive strength. where One G (x of ) is the perform the most important ance function; steps to and calculat x is the e p r(atndom v ) is to analyze ariable of the compre integral ssive of the strength probability . density function (PDF), leading to the diculty in probability computation. In this paper, a reliability p (t ) One of the most important steps to calculate is to analyze the integral of the probability analysis based on the TM method was proposed with a simpler calculation. density function (PDF), leading to the difficulty in probability computation. In this paper, a reliability The calculation steps of the time-varying TM reliability index ( (t)) [25] are 3M analysis based on the TM method was proposed with a simpler calculation. β (t ) 3M The calculation steps of the time-varying TM reli 2 ability index ( ) [25] are (t) = (3 9 + (t) 6 (t) (t))/ (t) (20) 2M 3M 3G 3G 3G β (t） = (3 − 9 + α （t） − 6α (t )β (t ) ) / α (t ) (20) 3M 3G 3G 2M 3G (t) = (t)/ (t) (21) 2M G G β (t） = μ (t ) / σ (t ) (21) 2 M G G (t) = (t)/(t) 18 (22) μ (t ) = α (t ) / λ (t ) − 18 (22) (t) = (t)/(t) (23) σ (t ) = [α (t ) − λ (t ) × 18] / α (t ) (23) = 2/ (t) (24) 3G α = 2 / α (t ) (24) where is the SM reliability index; is 3the G mean value of G(x); is the standard deviation of 2M G G G(x); and is the skewness of G(x). 3G μ σ β G(x) 2 M G G where is the SM reliability index; is the mean value of ; is the standard deviation of Figures 15 and 16 show the time-varying failure probability, time-varying reliability index, and the corr G(xesponding ) results of MC simulation G(x)( f = 18.0 N/mm ). Other failure probabilities and reliability 3G ; and is the skewness of . indices of dierent f can be developed using the same method. In this paper, an MC simulation was Figures 15–16 show the time-varying failure probability, time-varying reliability index, and the used to verify the accuracy of reliability. From the MC simulation, the errors of the failure probability corresponding results of MC simulation ( f = 18.0 N/mm ). Other failure probabilities and reliability and the TM reliability index were within 3%, as shown in Table 3. Obviously, from Figure 16, (t) 3M indices of different can be developed using the same method. In this paper, an MC simulation was used to verify the accuracy of reliability. From the MC simulation, the errors of the failure probability Appl. Sci. 2020, 10, 815 13 of 20 β (t ) 3 M and the TM reliability index were within 3%, as shown in Table 3. Obviously, from Figure 16, p (t) β (t ) β (t ) 2 M 3M had a better agreement than because skewness was considered in . Thus, and β (t ) 3M in this paper can be used to evaluate the structural safety with high accuracy. Table 3. Part of failure probabilities and reliability indices from 36 to 40 years. Error Error p (%) Serving year β 3M f 3M ( ) ( ) 36 7.69 0.09% 1.416 0.67% Appl. Sci. 2020, 10, 815 12 of 18 37 9.17 0.46% 1.320 0.09% 38 10.75 1.03% 1.229 0.42% 39 12.43 0.19% 1.142 0.92% had a better agreement than (t) because skewness was considered in (t). Thus, p (t) and (t) 2M 3M f 3M 40 14.19 0.53% 1.060 1.43% in this paper can be used to evaluate the structural safety with high accuracy. Appl. Sci. 2020, 10, 815 14 of 20 Figure 15. Time-varying failure probability and MC simulation ( f = 18.0 N/mm ). Figure 15. Time-varying failure probability and MC simulation ( f = 18.0 N/mm ). Figure 16. Time-varying reliability index and MC simulation ( f = 18.0 N/mm ). c 2 Figure 16. Time-varying reliability index and MC simulation ( ). f = 18.0 N/mm Table 3. Part of failure probabilities and reliability indices from 36 to 40 years. 7. Concrete Guaranteed Rate and Concrete Failure Time Serving Year p (%) Error (p ) Error ( ) 3M 3M f f On the basis of the building code calibration, concrete quality is determined by the guaranteed 36 7.69 0.09% 1.416 0.67% rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed 37 9.17 0.46% 1.320 0.09% 38 10.75 1.03% 1.229 0.42% rate was applied in this paper. The concrete failure time can be obtained when the qualified rate ( ) 39 12.43 0.19% 1.142 0.92% p p g q 40 14.19 0.53% 1.060 1.43% is equal to the guaranteed rate ( ). As shown in Figure 17, is represented by the dimension of the shaded area. μ = 22 V = 0.09 Figure 17. Normal distribution of and . is obtained through the probability degree (d) [39], which reflects the relative degree of limit error, as shown in Table 4. Appl. Sci. 2020, 10, 815 14 of 20 Figure 16. Time-varying reliability index and MC simulation ( f = 18.0 N/mm ). Appl. Sci. 2020, 10, 815 13 of 18 7. Concrete Guaranteed Rate and Concrete Failure Time 7. Concrete Guaranteed Rate and Concrete Failure Time On the basis of the building code calibration, concrete quality is determined by the guaranteed rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed On the basis of the building code calibration, concrete quality is determined by the guaranteed rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed rate was applied in this paper. The concrete failure time can be obtained when the qualified rate ( ) rate was applied in this paper. The concrete failure time can be obtained when the qualiﬁed rate (p ) is p p g q is equal to the guaranteed rate ( ). As shown in Figure 17, is represented by the dimension of equal to the guaranteed rate (p ). As shown in Figure 17, p is represented by the dimension of the g q the shaded area. shaded area. Figure 17. Normal distribution ofμ ==22 22 and V = 0.09. V = 0.09 Figure 17. Normal distribution of and . p is obtained through the probability degree (d) [39], which reﬂects the relative degree of limit is obtained through the probability degree (d) [39], which reflects the relative degree of limit error, as shown in Table 4. error, as shown in Table 4. Table 4. Probability degree and guaranteed rate. p (%) p (%) d d g g 1.35 91.15 1.65 95.05 1.40 91.92 1.70 95.54 1.45 92.65 1.75 95.99 1.50 93.32 1.80 96.64 1.55 93.94 1.85 96.78 1.60 94.52 1.90 97.13 Other probability degrees can be calculated by the linear interpolation. p (t) and d(t) are expressed as p (t) = 1 p (t) (25) d(t) = ((t) f )/(t) (26) (t) = (t)/(t) (27) where is the standard deviation. Figure 18 shows the relationship between p and p . From Figure 18, the failure time of q g f = 18.0 N/mm was expected to be 35.8 years. When (shown in Figure 14) was applied in the c 2 reliability calculation, the concrete failure time of f = 21.0 N/mm was 38.8 years, as shown in Figure 19. Appl. Sci. 2020, 10, 815 15 of 20 p (t ) d (t ) and are expressed as p (t ) = 1 − p (t ) (25) q f d (t ) = (μ (t ) − f ) / σ (t ) (26) (27) σ（t） = α (t) / λ (t) where is the standard deviation. Table 4. Probability degree and guaranteed rate. p p d g d g (%) (%) 1.35 91.15 1.65 95.05 1.40 91.92 1.70 95.54 1.45 92.65 1.75 95.99 1.50 93.32 1.80 96.64 1.55 93.94 1.85 96.78 1.60 94.52 1.90 97.13 Other probability degrees can be calculated by the linear interpolation. p p q g Figure 18 shows the relationship between and . From Figure 18, the failure time of f = 18.0 N/mm was expected to be 35.8 years. When (shown in Figure 14) was applied in the reliability calculation, the concrete failure time of was 38.8 years, as shown in Figure f = 21.0 N/mm Appl. Sci. 2020, 10, 815 14 of 18 Appl. Sci. 2020, 10, 815 16 of 20 Figure 18. Time-varying qualiﬁed rate and guaranteed rate of f = 18.0 N/mm . Figure 18. Time-varying qualified rate and guaranteed rate of f = 18.0 N/mm . Figure 19. Time-varying qualified rate and guaranteed rate of . Figure 19. Time-varying qualiﬁed rate and guaranteed rate of ff = =21.0 21.0N N/mm /mm . The failure probability and reliability index are equivalent indices of reliability. According to The failure probability and reliability index are equivalent indices of reliability. According to the the concrete failure time (35.8 years) calculated by the failure probability, the limit reliability index of concrete failure time (35.8 years) calculated by the failure probability, the limit reliability index of f = 18.0 N/mm was observed as 1.466. Other limit reliability indices of dierent f values can be c c f = 18.0 N/mm was observed as 1.466. Other limit reliability indices of different values can be observed through the same method. Failure mode II is deﬁned as the reliability index less than the observed through the same method. Failure mode II is defined as the reliability index less than the limit reliability index. limit reliability index. 8. Prediction Model of Equivalent Design Compressive Strength 8. Prediction Model of Equivalent Design Compressive Strength To test the accuracy of predicted results (concrete failure time) calculated by time-varying To test the accuracy of predicted results (concrete failure time) calculated by time-varying reliability, this paper applies the prediction model of equivalent design compressive strength ( f ). ce 0 0 By the deﬁnition of f , f f implies that p < p , at which point the concrete is regarded as out of c q g ce ce ce reliability, this paper applies the prediction model of equivalent design compressive strength ( ). service. McIntyre [41] calculated the equivalent design compressive strength from a set of core tested ' ' p ＜p f f ≤ f q g ce c ce By the de data by the finfollowing ition of steps: , implies that , at which point the concrete is regarded as 2 2 V = V 0.002 (28) out of service. McIntyre [41] calculated the equivalent design compressive strength from a set of core situ core in tested data by the following steps: 2 2 V = V − 0.002 (28) in core situ V = f V (29) in in situ， 30 situ — — (30) f = [1.724 − 4.138(V )] f ×1.08 in insitu ,17.5 situ， 30 c ,v (31) f = Rs × f c,v c (32) f = 1.481 f − 11 .24 ce insitu ,17 .5 in situ core where and are the coefficients of variation of the situ compressive strength and the core in situ， 30 test strength, respectively; is the converting coefficient of variation when there are fewer c,v than 30 tests; is the multiplying factor (see [41]); is the mean value of core strength (with Appl. Sci. 2020, 10, 815 15 of 18 V = f V (29) situ,30 situ in in f = [1.724 4.138(V )] f 1.08 (30) insitu,17.5 situ,30 c,v in f = Rs f (31) c,v c f = 1.481 f 11.24 (32) ce insitu,17.5 Appl. Sci. 2020, 10, 815 17 of 20 where V and V are the coecients of variation of the situ compressive strength and the core situ core in test strength, respectively; V is the converting coecient of variation when there are fewer than situ,30 in insitu ,17.5 core coefficient of variation = ); is the mean value of situ compressive strength (with 30 tests; f is the multiplying factor (see [41]); f is the mean value of core strength (with coecient of c,v variation = V ); f is the mean value ce of situ compressive strength (with coecient of variation coefficient ofcore variat insitu ion = 1 ,17.57.5%); and is the equivalent design strength. = 17.5%); and f is the equivalent design strength. ce core The in this paper is expressed as Equation (33) based on Figure 8. The V in this paper is expressed as Equation (33) based on Figure 8. core V (t ) = 0.07 × exp(0.033t ) (33) core V (t) = 0.07 exp(0.033t) (33) core The relative equivalent design compressive strength is expressed as The relative equivalent design compressive strength is expressed as R = f / f (34) f ce c R = f / f . (34) f ce R ＜ 1 Failure mode III occurs when . Based on Equations (28)–(34), failure mode III was reached Failure mode III occurs when R < 1. Based on Equations (28)–(34), failure mode III was reached 2 2 at 36.7 years when 2, and at 38.3 years when 2 (see Figure 20). These f = 18.0 N/mm f = 21.0 N/mm at 36.7 years when f = c 18.0 N/mm , and at 38.3 years when f = c 21.0 N/mm (see Figure 20). These c c results were close to the calculated results of 35.8 years in Figure 18, and 38.8 years in Figure 19, results were close to the calculated results of 35.8 years in Figure 18, and 38.8 years in Figure 19, respectively. The calculated results were in good agreement because both the mean and coefficient respectively. The calculated results were in good agreement because both the mean and coecient of variation were considered in the two prediction models. Moreover, the failure modes of the two of variation were considered in the two prediction models. Moreover, the failure modes of the two models were based on the concrete guaranteed rate. models were based on the concrete guaranteed rate. 2 2 22 Figure 20. Relative equivalent design concrete strength ( f = 18.0 N/mm and f = 21.0 N/mm ). Figure 20. Relative equivalent design concrete strength (cf = 18.0 N/mm and cf = 21.0 N/mm ). c c 9. Example 9. Example f of beam I and beam II of dierent buildings in Yokohama, Japan, was 18.0N/mm , and the c 18.0N / mm service years of beam of beam Ι and bea I and m beam II of different II were 31 buildings and 38 years, in Yokoha respectively ma, Jap . an, was , and the service years of beam Ι and beam II were 31 and 38 years, respectively. In this example, f , t , and t were 18.0 N/mm , 31 years, and 38 years, respectively. Climate c 1 2 statistics were collected on a related weather website. f t t 1 2 18. 0 N/mm In this example, , , and were , 31 years, and 38 years, respectively. Climate statistics were collected on a related weather website. 9.1. Calculation Ι Using Equations (4) and (10), the relative compressive strengths of beam Ι and beam II were calculated as Rs = 1.36 Rs = 1.34 , . 2 ,Ⅰ 2 , Ⅱ According to failure mode Ι, both beams were still in service. Appl. Sci. 2020, 10, 815 16 of 18 9.1. Calculation I Using Equations (4) and (10), the relative compressive strengths of beam I and beam II were calculated as Rs = 1.36, Rs = 1.34. 2, 2, According to failure mode I, both beams were still in service. 9.2. Calculation II Using Equations (14) and (15) and (20)–(24), the corresponding reliability indices were calculated as = 1.9787, = 1.2288. 3M 3M , , According to failure mode II, beam I was in service, whereas beam II had been destroyed. 9.3. Calculation III Using Equations (4), (10), and (28)–(34), the corresponding equivalent design compressive strengths are calculated as R = 1.395, R = 0.9. f 2, f 2, According to failure mode III, the result of the safety evaluation coincided with that obtained in calculation II. On the basis of the analysis and discussion above, the result of Calculation I is dierent from others because only the mean value is considered. Therefore, the ﬁrst result is inconvincible. From the second and third results, beam I is relatively secure. However, beam II is suggested to be retroﬁtted. It is suggested that the member of failure probability of less than 0.5 is supposed to be replaced rather than strengthened. 10. Conclusions (1) The environmental factors (temperature and humidity) are considered in the prediction model of MVCS. (2) Compared with other distributions, such as a normal distribution and lognormal distribution, the gamma distribution is applied to describe the distribution of compressive strength with the highest R-square. (3) The time-varying TM reliability, with higher accuracy in contrast to SM method, is proposed to predict the concrete failure time. (4) Compared with the results of MC simulation, the prediction model of time-varying reliability based on the TM method is accurate, and the errors are within 3%. (5) Tested by the model for calculating an equivalent design compressive strength, the errors of the concrete failure time calculated by time-varying reliability are acceptable. Author Contributions: conceptualization, Y.Z. and J.W.; methodology, J.W.; software, W.Z.; validation, J.W. and W.Z.; formal analysis, W.Z.; investigation, W.Z.; resources, X.Z.; data curation, J.W. and Y.Z.; writing—original draft preparation, W.Z.; writing—review and editing, W.Z.; visualization, W.Z.; supervision, J.W.; project administration, Y.Z. and X.Z.; funding acquisition, Y.Z. and X.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research work was supported by the Natural Science Foundation of China (Grant No. 51878413). Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Wahid, F.; Yu, B.; Tuan, D.N.; Allan, M.; Priyan, N. New advancements, challenges and opportunities of multi-storey modular buildings—A state-of-the-art review. Eng. Struct. 2019, 183, 883–893. Appl. Sci. 2020, 10, 815 17 of 18 2. Lantsoght, E.O.; van der Veen, C.; de Boer, A.; Hordijk, D.A. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/prediction-of-concrete-failure-time-based-on-statistical-properties-of-mcDS0CYvDz

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Abstract

applied sciences Article Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength 1 1 , 2 3 Jiao Wang , Weiji Zheng * , Yangang Zhao and Xiaogang Zhang School of Civil Engineering, Guangzhou University, Guangzhou 510006, China; cewangjiao@e.gzhu.edu.cn Department of Architecture and Building Engineering, Kanagawa University, Yokohama 2218686, Japan; zhao@kanagawa-u.ac.jp College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China; szzxg@szu.edu.cn * Correspondence: 2111816256@e.gzhu.edu.cn Received: 15 December 2019; Accepted: 21 January 2020; Published: 23 January 2020 Abstract: Since the heterogeneity of the cement-based material contributes to a random spatial distribution of compressive strength, a reliability analysis based on the compressive strength of concrete is fundamental to carry out structural safety assessment. By analyzing 10,317 datapoints on compressive strength of concrete, a time-varying reliability evaluation based on the third-moment (TM) method was proposed to predict the service life of concrete. Unlike the second-moment (SM) method, skewness is taken into account in the TM; thus, the calculated result of concrete failure time based on the TM is more accurate. In this paper, the errors of the calculated results using time-varying reliability evaluation are within 3%, as shown by Monte Carlo (MC) simulation. In addition, the proposed model (aiming to calculate the equivalent design compressive strength) veriﬁes the concrete failure time calculated by time-varying reliability. According to the results, concrete failure times calculated by these two models are in good agreement. Overall, based on the simple and eective methodology adopted in this paper, it is feasible to develop time-varying reliability based on other factors that might also lead to concrete failure, such as carbonation-induced corrosion, cracking, or deﬂection of the concrete. Keywords: concrete strength; failure probability; third-moment method; equivalent design concrete strength 1. Introduction Reinforced concretes have been applied worldwide in structural engineering since the beginning of the nineteenth century [1–5]. However, structure damage has increasingly occurred under aggressive service environments, which might be caused by ambient temperature, freeze–thaw cycles, precipitation, carbon dioxide, chloride, and sulfate [6–8]. Therefore, it is signiﬁcant to implement eective strategies to guarantee structural safety and strengthen vulnerable buildings. In this paper, a proposed model based on the compressive strength of concrete was applied to predict the concrete failure time. Carbonation is one of the most fundamental triggers of concrete deterioration [9–13], and carbonation depth is used to describe the level of carbonation-induced corrosion. As presented in the existing paper, the carbonation depth has both favorable and unfavorable eects on the compressive strength of concrete. On the positive side, carbonation-induced corrosion contributes to the increase of the compressive strength and the harnessing of concrete because numerous concrete pores are occupied by the carbonation product, calcium carbonate [14]. On the negative side, too much calcium carbonate that exceeds the limit of concrete pores might cause additional internal pressure and Appl. Sci. 2020, 10, 815; doi:10.3390/app10030815 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 815 2 of 18 micro-cracking [15,16]. In this paper, the relationship between carbonation depth and the compressive strength of concrete is analyzed using a prediction model of carbonation depth [17]. Since the presence of variations associated with climatic factors (e.g., temperature and humidity), concrete materials, and geometric conﬁgurations [18,19], the compressive strength of concrete should be regarded as a variable in order to account for these uncertainties in deterioration. In practical engineering problems, these unavoidable uncertainties are often qualiﬁed by probabilistic models [20–22], among which reliability analyses are widely applied in structural safety assessments [23–25]. In the existing time-varying reliability models, the probabilities of corrosion initiation and corrosion damage are calculated based on the change in climatic variables [26]. Szilágyi [27] developed a phenomenological constitutive model for obtaining the rebound hardness of a surface concrete as a property of the time-dependent material. Nevertheless, this model has not yet been ratiﬁed for practical applications. A previously conducted study [28] exhibits the property of compressive strength in the case of deteriorating concrete in the Tohoku region of Japan; however, the prediction model of compressive strength and the method of structural safety evaluation were not provided. Generally, most existing probabilistic models may not be suitable for evaluating the service life of concrete. Based on a large number of tested data on compressive strength, an eective model based on the reliability to evaluate the service life of concrete was developed. It is noted that dierent sizes and shapes of the tested components aect the compressive strength of concrete. Therefore, the conversion and standardization of the value of compressive strength may produce an inaccurate result [29]. In this paper, all the shapes and sizes of the experimental members were uniﬁed. Additionally, the compressive strengths of all the samples were tested using core pulling. Gao [30] proposed that the compressive strength obeys a normal distribution; however, by analyzing 10,317 datapoints considered in this study, it is observed that a gamma distribution is more appropriate to describe the distribution of compressive strength. Moreover, though Gao’s model serves the purpose of evaluating the compressive strength, only the mean value of compressive strength is considered. Therefore, the probabilistic model may not obtain the precise prediction of concrete failure time. Regarding the discreteness of the statistic, both mean value and coecient of variation are expected to be considered in the model to evaluate the service life of concrete. Thus, a time-varying evaluation model based on reliability was proposed in this study. The failure probability and reliability index equivalently evaluate the structural safety. However, the reliability index, being very simple and with respect to function, is not integral, which is in contrast to the calculation of failure probability. Among all the methods for computing the reliability index, moment methods are used worldwide to observe the reliability index [25]. Compared with the second-moment (SM) method, the reliability evaluation based on the third-moment (TM) method is more accurate because the skewness (the third dimensionless central moment) is considered [31,32]. Tested by MC simulation, the results evaluated by the TM reliability are more accurate than those by the SM reliability. In this paper, the concrete guaranteed rate was applied to predict the concrete failure time calculated by time-varying reliability. To testify the reliability-based computation of the concrete failure time, the equivalent design compressive strength (functioning as the ultimate concrete compressive strength) was investigated to observe the concrete failure time. The concrete failure times computed by both models agree well. 2. Distribution Fitting of the Compressive Strength of Concrete Deterioration in concrete construction is a severe problem that leads to durability problems and even structural failure. In order to prevent concrete failure and retroﬁt the aging constructions over time, periodical testings of the structural components are conducted every three years, as shown in Figure 1. In this paper, all data were obtained from these testings, and all tested components were of natural ﬁne aggregated concrete, taken from 1061 buildings (e.g., schools, residences, and oce buildings) in Yokohama, Japan, where the average temperature and humidity were 15.208 C and Appl. Sci. 2020, 10, 815 3 of 20 Appl. Sci. 2020, 10, 815 3 of 18 Appl. Sci. 2020, 10, 815 3 of 20 core compressive strength ( ), and the average dimensions of the components were 80 mm in diameter Appl. and Sci. 10 2020 0 mm in hei , 10, 815 ght. 3 of 20 core compressive strength ( ), and the average dimensions of the components were 80 mm in diameter 69%, respectively. The tested compressive strength in these experiments was regarded as the core and 100 mm in height. compressive strength ( f ), and the average dimensions of the components were 80 mm in diameter cor core e compressive strength ( ), and the average dimensions of the components were 80 mm in diameter and 100 mm in height. and 100 mm in height. Figure 1. The compressive strength experiment of concrete core. The tested data (10,317) were divided into several groups according to different design values Figure 1. The compressive strength experiment of concrete core. of concrete strength ( Figure 1. Figure ) and t 1. The c The he compr o service mpressive essive tim str str e of e ength ng concr th ex experiment periment of ete (yearof s),concrete concr as sh ete own cor core. in Fig e. ures 2 and 3. The tested data (10,317) were divided into several groups according to different design values The tested data (10,317) c were divided into several groups according to dierent design values of The tested data (10,317) were divided into several groups according to different design values of concrete strength ( ) and the service time of concrete (years), as shown in Figures 2 and 3. concrete strength ( f ) and the service time of concrete (years), as shown in Figures 2 and 3. c f of concrete strength ( ) and the service time of concrete (years), as shown in Figures 2 and 3. f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). Figure 2. Statistics of concrete strength ( f = 18.0N/mm ). c 2 f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). f = 18.0 N/mm Figure 2. Statistics of concrete strength ( ). Figure 3. Statistics of concrete strength ( f = 21.0N/mm ). Appl. Sci. 2020, 10, 815 4 of 20 Appl. Sci. 2020, 10, 815 4 of 20 f = 21.0 N/mm c 2 Figure 3. Statistics of concrete strength ( ). f = 21.0 N/mm Appl. Sci. 2020, 10, 815 c 4 of 18 Figure 3. Statistics of concrete strength ( ). MATLAB software was used to analyze the fitting distributions of datasets with the sample MATLAB software was used to analyze the fitting distributions of datasets with the sample significanc MATLAB e level a = softwar 0.05. e was The used tests oto f ganalyze amma distri the b ﬁtting ution, n distributions ormal distribution of datasets , Rayle with igh distribution, the sample significance level a = 0.05. The tests of gamma distribution, normal distribution, Rayleigh distribution, signiﬁcance and extreme level value d a = 0.05. istribThe ution wer tests of e e gamma xecuted b distribution, ased on these da normal distribution, ta. It was foRaylei und tha ght the distribution, gamma and extreme value distribution were executed based on these data. It was found that the gamma and distri extr buti eme on ha value d the hi distribution ghest goodness of were executed fit, as sh based own in onFigure 4. Figure 5 these data. It was shows the found that comparison the gamma of distribution had the highest goodness of fit, as shown in Figure 4. Figure 5 shows the comparison of distribution R-squares bet had ween the gamma distr the highest goodness ibution and the of ﬁt, as shown noin rmal d Figuristr e 4i.bution of d Figure 5 shows ifferent service the comparison years. R-squares between the gamma distribution and the normal distribution of different service years. of Accordi R-squar ng to Fi es between gure 5, the ga the gamma mma distribution distributi and on proved to be more a the normal distributionppl ofidi cabl ereent to describe the service years. According to Figure 5, the gamma distribution proved to be more applicable to describe the Accor compressi dingve to Figur strength. e 5, the Addi gamma tionadistribution lly, for a deteri proved orating proce to be moress, applicable the stoch to astic describe gamma proce the comprss w essive as compressive strength. Additionally, for a deteriorating process, the stochastic gamma process was adopted to analyze the time-varying reliability. Abdel-Hameed [33] was the first to propose the strength. Additionally, for a deteriorating process, the stochastic gamma process was adopted to adopted to analyze the time-varying reliability. Abdel-Hameed [33] was the first to propose the analyze gamma process the time-varying as the first reliability choice to descri . Abdel-Hameed be the pe[rfor 33] ma was ncthe e dﬁrst egrad to atpr ioopose n procthe ess [34]. Mo gamma pr reover, ocess gamma process as the first choice to describe the performance degradation process [34]. Moreover, van Noortwijk modeled the stochastic process of deterioration with a gamma process, which had as the ﬁrst choice to describe the performance degradation process [34]. Moreover, van Noortwijk van Noortwijk modeled the stochastic process of deterioration with a gamma process, which had modeled independent the gamma-d stochastic istributed inc process of deterioration rements with ide with ntia cagamma l scale pa pr raocess, meters [35 which ]. The resul had independent ts indicate independent gamma-distributed increments with identical scale parameters [35]. The results indicate that a gamma process can properly model the monotonic behavior of aging. Overall, the preceding gamma-distributed increments with identical scale parameters [35]. The results indicate that a gamma that a gamma process can properly model the monotonic behavior of aging. Overall, the preceding pr rese ocess arch can figpr ure shows operly model that t the he gamma monotonic process behav is ior s ofuit aging. able tOverall, o model the gra pr du eceding al and monot research onica ﬁgur lly e research figure shows that the gamma process is suitable to model gradual and monotonically accumulating damage over time [35,36]. Based on these existing conclusions and the analysis in this shows that the gamma process is suitable to model gradual and monotonically accumulating damage accumulating damage over time [35,36]. Based on these existing conclusions and the analysis in this over paper, the time [ gamma 35,36]. distribution is a Based on theseppropria existing tconclusions e to describe the d and the istribution of co analysis in this mpressive paper, the stregamma ngth. paper, the gamma distribution is appropriate to describe the distribution of compressive strength. distribution is appropriate to describe the distribution of compressive strength. Figure 4. Distribution fitting of compressive strength. Figure 4. Distribution ﬁtting of compressive strength. Figure 4. Distribution fitting of compressive strength. Gamma Gamma 1.02 1.02 Normal Normal 0.98 0.98 0.96 0.96 0.94 0.94 0.92 0.92 0.9 0.9 0.88 0.88 Time (year) 35 36 37 38 39 40 41 42 43 44 Time (year) 35 36 37 38 39 40 41 42 43 44 Figure 5. Contrast of goodness of ﬁt for gamma distribution and normal distribution. Figure 5. Contrast of goodness of fit for gamma distribution and normal distribution. Figure 5. Contrast of goodness of fit for gamma distribution and normal distribution. 3. Prediction for the Mean Value of Compressive Strength 3. Prediction for the Mean Value of Compressive Strength R-square R-square Appl. Sci. 2020, 10, 815 5 of 18 3. Prediction for the Mean Value of Compressive Strength Carbonation-induced corrosion can lead to a residual capacity loss of the reinforced concrete structures [37]. Generally, carbonation depth is used to deﬁne the carbonized level of the concrete. To analyze the relation between the carbonation depth and the compressive strength of concrete, and to develop a prediction model for the mean value of compressive strength (MVCS) in which the environment factors (temperature and humidity) are considered, the ratio of compressive strength to carbonation depth was investigated. In addition, the concrete failure time calculated by this prediction model was used to compare with those calculated by time-varying reliability. Based on the existing model of carbonation depth [17], the carbonation depth is expressed as 0.05 f cm D(t) = 7.742e t (1) T RH 0.25 = (T/T ) (2) T 0 = RH(100 RH)/RH (100 RH ) (3) RH 0 0 f = f /(1 1.645) (4) cm cu,k where D is the mean value of carbonation depth in a nonstandard condition (mm); is the coecient of temperature; is the coecient of humidity; T is the environmental temperature ( C); T is the RH 0 average temperature ( C); RH is the environmental humidity (%); RH is the average humidity (%); t is the service time of concrete (year); f is the mean value of the standard value of compressive strength cm 2 2 (N/mm ); f (equal to concrete grade) is the standard value of compressive strength (N/mm ), cu,k which is observed by converting f (design value of compressive strength (N/mm )) [38]; and is the coecient of variation, which can be obtained via the Code for Design of Concrete Structures [39]. According to the investigation of the relevant climate data, the climate changes among dierent years were not apparent. In order to simplify the calculation of Equation (1), T is set as the median of all mean values of the temperature of dierent years, and RH is the median of all mean values of the humidity of dierent years. Dierent from the prediction model of carbonation depth in the existing paper [17], f is replaced by f , which is more appropriate to calculate the ratio Ra(t), as shown in cu,k cm Equation (4). Relevant parameters calculating the carbonation depth are presented in Table 1. Table 1. Relevant parameters calculating the carbonation depth in Yokohama, Japan. 2 2 Relevant Parameter f =18.0 N/mm f =21.0 N/mm c c T( C) 15.425 15.425 T ( C) 15.208 15.208 RH(%) 62 62 RH (%) 69 69 f (N/mm ) 37.6 45 cu,k 0.16 0.15 f (N/mm ) 51 59.8 cm To analyze the relation between compressive strength of concrete and carbonation depth, the ratio of relative concrete strength to carbonation depth is developed as Ra(t) = Rs(t)/D(t) (5) Rs(t) = (t)/ f (6) Rs c where Ra is the ratio of the relative MVCS to the mean value of carbonation depth (MVCD); Rs is the relative MVCS; and is the MVCS of dierent years (N/mm ), which is obtained based on 10,317 Rs experimental datapoints, as shown in Table 2. Appl. Sci. 2020, 10, 815 6 of 20 Rs(t ) = μ (t ) / f (6) Rs c Ra Rs where is the ratio of the relative MVCS to the mean value of carbonation depth (MVCD); is Rs N / mm the relative MVCS; and is the MVCS of different years ( ), which is obtained based on 10,317 experimental datapoints, as shown in Table 2. 2 2 f = 21. 0 N/mm f = 18. 0 N/mm c c Table 2. Mean value of compressive strength with (a) and (b) . Appl. Sci. 2020, 10, 815 6 of 18 (a) Mean Value of Compressive Strength with f = 21.0 N/mm 2 2 Time (Year) 35 36 37 38 39 40 41 42 43 44 45 Table 2. Mean value of compressive strength with (a) f = 21.0 N/mm and (b) f = 18.0 N/mm . c c Rs1 31.95 3 (a)0 Mean .70 V3 alue0.19 of Compressive 31.29 28Strength .04 25with .27 2 f =3.78 21.0 N/2 mm3.71 23.54 24.51 27.43 N/ mm ( ) Time (Year) 35 36 37 38 39 40 41 42 43 44 45 (b) Mean Value of Compressive Strength with f = 18.0 N/mm 2 c (N/mm ) 31.95 30.70 30.19 31.29 28.04 25.27 23.78 23.71 23.54 24.51 27.43 Rs1 Time (Year) 42 43 44 45 46 47 48 49 50 51 52 (b) Mean Value of Compressive Strength with f = 18.0 N/mm Rs Time (Year) 42 43 44 45 46 47 48 49 50 51 52 22.65 23.00 24.15 23.34 23.14 22.96 22.88 21.54 22.47 22.19 19.73 N / mm 2 ( ) 22.65 23.00 24.15 23.34 23.14 22.96 22.88 21.54 22.47 22.19 19.73 (N/mm ) Rs2 Time (year) 53 54 Time (year) 53 54 μ 2 (N/mm ) 20.77 20.03 Rs2 Rs2 20.77 20.03 N / mm ( ) According to Table 2, it was found that the compressive strength of concrete decreased with the According to Table 2, it was found that the compressive strength of concrete decreased with the increase of time. increase of time. Based on Equations (1)–(6) and Tables 1 and 2, Ra values of dierent servicing years were ﬁtted as Based on Equations (1)–(6) and Tables 1 and 2, Ra values of different servicing years were fitted shown in Figure 6. as shown in Figure 6. Figure 6. Fitting of the ratio of the relative concrete strength to the mean value of carbonation depth. Figure 6. Fitting of the ratio of the relative concrete strength to the mean value of carbonation depth. According to Figure 6, Ra decreased as time increased, showing that the compressive strength According to Figure 6, Ra decreased as time increased, showing that the compressive strength of concrete decreased with the increase of carbonation depth over time. In addition, it is known that of concrete decreased with the increase of carbonation depth over time. In addition, it is known that the carbonation depth increased with the increase of time. Therefore, combined with the deﬁnition of the carbonation depth increased with the increase of time. Therefore, combined with the definition carbonation depth, it was concluded that carbonation-induced corrosion can result in the decrease of of carbonation depth, it was concluded that carbonation-induced corrosion can result in the decrease compressive strength of concrete, which agrees with the conclusion drawn from Zha [40]. of compressive strength of concrete, which agrees with the conclusion drawn from Zha [40]. The ﬁtting curves in Figure 6 are expressed as The fitting curves in Figure 6 are expressed as Ra (t) = 0.02t + 1.279 (7) Ra (t) = 0.0058t + 0.5445 (8) where Ra is the ratio of the relative compressive strength to carbonation depth with f = 21.0 N/mm ; and Ra is the ratio of the relative compressive strength to carbonation depth with f = 18.0 N/mm . 2 c By comparing the changing rate of Ra (t) with Ra (t), the deterioration rate of concrete with 2 2 f = 21.0 N/mm is faster than that of f = 18.0 N/mm . c c Appl. Sci. 2020, 10, 815 7 of 20 Ra (t ) = −0.02t + 1.279 (7) Ra (t ) = −0.0058t + 0.5445 (8) Ra where is the ratio of the relative compressive strength to carbonation depth with Ra ; and is the ratio of the relative compressive strength to carbonation depth with f = 21.0 N/mm f = 18.0 N/mm . Appl. Sci. 2020, 10, 815 7 of 18 Ra (t ) Ra (t) 1 2 By comparing the changing rate of with , the deterioration rate of concrete 2 2 with f = 21.0 N/mm is faster than that of f = 18.0 N/mm . Onc the basis of Equations (1), (5), (7) and c (8), the new prediction models of Rs are proposed as Rs On the basis of Equations (1), (5), (7) and (8), the new prediction models of are proposed as 0.05 f cm1 Rs (t) = D (t) Ra (t) = 7.742e t (0.02t + 1.279) (9) 1 1 1 T RH __ − 0.05 f cm 1 (9) Rs (t ) = D (t ) × Ra (t ) = 7.742 e × t × α × α × (−0.02t + 1.279 ) 1 1 T RH 0.05 f cm2 Rs (t) = D (t) Ra (t) = 7.742e t (0.0058t + 0.5445) (10) 2 2 2 T RH __ − 0.05 f where Rs is the relative compressive strength of cm 2 f = 21.0 N/mm ; Rs is the relative compressive (10) 1 c 2 Rs (t ) = D (t ) × Ra (t ) = 7.742e × t × α × α × (−0.0058t + 0.5445) 2 2 T RH strength of f = 18.0 N/mm ; f is the mean value of the standard value of compressive strength cm1 Rs Rs with f = 21.0 N/mm ; and f is the mean value of the standard value of compressive strength with 1 2 where c is the relative com cmp 2 ressive strength of ; is the relative compressive f = 21.0 N/mm f = 18.0 N/mm . 2 cm 1 strength of ; is the mean value of the standard value of compressive strength f = 18.0 N/mm Figure 7 is c developed from Equations (9) and (10), showing that with an increase of service time, the compressive strength decreased, and the speed of strength deterioration increased. Rs(t) = 1.0 is cm 2 2 with ; and is the mean value of the standard value of compressive strength with f = 21.0 N/mm the limit state of concrete failure in conducting concrete safety assessments (failure mode I). As shown f = 18.0 N/mm . c 2 in Figure 7, the concrete failure time of f = 21.0 N/mm was observed to be 48.2 years, while that Figure 7 is developed from Equations (9) and (10), showing that with an increase of service time, of f = 18.0 N/mm was observed to be 61.8 years. However, the coecient of variation was not the compressive strength decreased, and the speed of strength deterioration increased. considered in this model, leading to the inaccurate prediction of concrete failure time. Rs (t ) = 1 . 0 Figures 8–10 also present the corresponding ﬁtting curves with acceptable R-squares, indicating is the limit state of concrete failure in conducting concrete safety assessments (failure that the inﬂuence of f on the coecient of variation, the skewness, and the kurtosis was negligible. mode Ι). As shown in Figure 7, the concrete failure time of was observed to be 48.2 f = 21.0 N/mm Based on Figures 8–10, three eigenvalues increased with the increase of time. Besides, the coecient of years, while that of f = 18.0 N/mm was observed to be 61.8 years. However, the coefficient of variation increased over time, resulting in dispersion expansion of the statistic. Therefore, the accuracy variation was not considered in this model, leading to the inaccurate prediction of concrete failure of the observed concrete failure time determined only by the mean value would decrease when time. time increases. 2 2 Figure 7. Relative concrete strength ( f = 18.0N/ mm and f = 21.0 N/mm ). c c 2 2 Figure 7. Relative concrete strength ( f = 18.0 N/mm and f = 21.0 N/mm ). c c 4. Eect of f on the Distribution of Compressive Strength Based on the analysis of distribution ﬁtting, a gamma distribution was used to describe the 4. Effect of on the Distribution of Compressive Strength distribution of compressive strength. To investigate the eect of f on the distribution, the coecient of variation, skewness, and kurtosis of tested data with dierent f values were investigated, as presented in Figures 8–10. Appl. Sci. 2020, 10, 815 8 of 20 Appl. Sci. 2020, 10, 815 8 of 20 Based on the analysis of distribution fitting, a gamma distribution was used to describe the Based on the analysis of distribution fitting, a gamma distribution was used to describe the distribution of compressive strength. To investigate the effect of on the distribution, the distribution of compressive strength. To investigate the effect of on the distribution, the coefficient of variation, skewness, and kurtosis of tested data with different values were investigated, as presented in Figures 8–10. coefficient of variation, skewness, and kurtosis of tested data with different values were Appl. Sci. 2020, 10, 815 8 of 18 investigated, as presented in Figures 8–10. Figure 8. Time-varying coefficient of variation. Figure Figure 8. 8. T Time- ime-varying varying coefficient of coecient of variation. variation. Appl. Sci. 2020, 10, 815 9 of 20 Figure 9. Time-varying skewness. Figure 9. Time-varying skewness. Figure 9. Time-varying skewness. Figure 10. Time-varying kurtosis. Figure 10. Time-varying kurtosis. Figures 8–10 also present the corresponding fitting curves with acceptable R-squares, indicating that the influence of on the coefficient of variation, the skewness, and the kurtosis was negligible. Based on Figures 8–10, three eigenvalues increased with the increase of time. Besides, the coefficient of variation increased over time, resulting in dispersion expansion of the statistic. Therefore, the accuracy of the observed concrete failure time determined only by the mean value would decrease when time increases. According to the property of gamma distribution, these three eigenvalues can be calculated by the shape parameter , which are expressed as α λ 1 V = × = (11) λ α α = (12) α = (13) α α V 3 4 where is the coefficient of variation; is the shape parameter; is the skewness; and is the kurtosis. Since the three eigenvalues are determined only by , it is concluded that also has no influence on . 5. Scale Parameter and Shape Parameter of Gamma Distribution A reliability analysis was applied in this paper to predict the concrete failure time with high accuracy. In order to develop a time-varying reliability, the parameters ( and ) of the gamma α λ distribution function at different service years need to be investigated. Different from , varies with different values of . The parameters and the corresponding fitting curves are observed in Figures 11 and 12. Appl. Sci. 2020, 10, 815 9 of 18 According to the property of gamma distribution, these three eigenvalues can be calculated by the shape parameter , which are expressed as V = = p (11) = (12) = (13) where V is the coecient of variation; is the shape parameter; is the skewness; and is the kurtosis. Since the three eigenvalues are determined only by , it is concluded that f also has no inﬂuence on . 5. Scale Parameter and Shape Parameter of Gamma Distribution A reliability analysis was applied in this paper to predict the concrete failure time with high accuracy. In order to develop a time-varying reliability, the parameters ( and ) of the gamma Appl. Sci. 2020, 10, 815 10 of 20 distribution function at dierent service years need to be investigated. Dierent from , varies with dierent values of f . The parameters and the corresponding ﬁtting curves are observed in Figures 11 Appl. Sci. 2020, 10, 815 10 of 20 and 12. Figure 11. Scale parameter ( ) fitting ( ). f = 18.0 N/mm Figure 11. Scale parameter ( ) ﬁtting ( f = 18.0 N/mm ). λ 1 Figure 11. Scale parameter ( ) fitting ( ). f = 18.0 N/mm Figure 12. Shape parameter () ﬁtting. Figure 12. Shape parameter ( ) fitting. Figure 12. Shape parameter ( ) fitting. Equations (14) and (15) are the predicted models of these two parameters obtained by curve fitting. Equations (14) and (15) are the predicted models of these two parameters obtained by curve λ (t ) = 12 .94 × exp (−0.072t ) (14) fitting. λ α ( (tt ) ) = = 12 428.4 .94 ××exp exp(− ( − 00 .072 .08tt ) ) (1 (14) 5) Another method for computing the time-varying scale parameter is proposed without fitting α (t ) = 428.4 × exp ( −0.08t ) (15) λ (t ) . The proposed method is presented in the subsequent section. Another method for computing the time-varying scale parameter is proposed without fitting In this paper, MVCS was computed by Equation (16-a). Besides, based on the property of gamma λ (t ) . The proposed method is presented in the subsequent section. distribution, MVCS can also be expressed as Equation (16-b). In this paper, MVCS was computed by Equation (16-a). Besides, based on the property of gamma μ (t) = Rs (t) × f distribution, MVCS can also be expressed as Equation (16-b). (16-a) Rs n c μ (t) = Rs (t) × f (16-a) Rs n c μ (t ) = α (t ) / λ (t ) (16-b) g n μ (t ) = α (t ) / λ (t ) μ μ (16-b) g n Rs where and are both the mean value of compressive strength; and n is used to distinguish Rs λ μ μ c and Rs with different . where and are both the mean value of compressive strength; and n is used to distinguish f μ Rs λ c μ g and with different . Rs Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed based on Equations (10), (16-a) and (14), (15), (16-b), respectively. The difference between μ g Rs Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed based on Equations (10), (16-a) and (14), (15), (16-b), respectively. The difference between Appl. Sci. 2020, 10, 815 10 of 18 Equations (14) and (15) are the predicted models of these two parameters obtained by curve ﬁtting. (t) = 12.94 exp(0.072t) (14) (t) = 428.4 exp(0.08t) (15) Another method for computing the time-varying scale parameter is proposed without ﬁtting (t). The proposed method is presented in the subsequent section. In this paper, MVCS was computed by Equation (16a). Besides, based on the property of gamma distribution, MVCS can also be expressed as Equation (16b). (t) = Rs (t) f (16a) Rs n c (t) = (t)/ (t) (16b) g n where and are both the mean value of compressive strength; and n is used to distinguish Rs and Rs g with dierent f . Appl. Sci. 2020, 10, 815 c 11 of 20 Figure 13 shows two prediction models of and of f = 18.0 N/mm , which are expressed Rs g c based on Equations (10), (16a) and (14), (15), (16b), respectively. The dierence between the predicted Rs the predicted results exists because and were both obtained based on curve fitting. The results exists because and were both obtained based on curve ﬁtting. The errors between the Rs g errors between the predicted results were less than 2%. predicted results were less than 2%. Figure 13. Comparison of results calculated by and ( f = 18.0 N/mm ). g c Rs μ g Rs Figure 13. Comparison of results calculated by and ( f = 18.0 N/mm ). According to Figure 13, it is concluded that and were equivalent to evaluate the compressive Rs g strength, which is expressed as Equation (17a). Therefore, an approximate expression, formulated based on Equation (17a) for the time-varying scale parameter, is elaborated as presented by Equation (17b). Rs According to Figure 13, it is concluded that and were equivalent to evaluate the compressive strength, which is expressed as Equation (17-a). Therefore, an approximate expression, (t) formulated based on Equation (17-a) forRs the t (ti)me-v f ar =ying scale parameter, is elaborated as present (17a) ed n c (t) by Equation (17-b). (t) α (t) (t) = (17b) Rs (t) ×Rs f =(t) f n c (17-a) n c λ (t) In light of Equation (17b), it is feasible to obtain without curve ﬁtting. By using the proposed method, of f = 21.0 N/mm is obtained based on Equations (9), (15), and (17b), as shown in α (t) λ (t) = Figure 14. (17-b) Rs (t) × f n c In light of Equation (17-b), it is feasible to obtain without curve fitting. By using the proposed method, of is obtained based on Equations (9), (15), and (17-b), as shown in f = 21.0 N/mm Figure 14. Appl. Sci. 2020, 10, 815 11 of 18 Appl. Sci. 2020, 10, 815 12 of 20 Figure 14. Scale parameter ( ) with f f = =21.0N 21.0N//mm mm . 2 2 c c Figure 14. Scale parameter ( ) with . 6. Time-Varying Reliability Evaluation Based on the Tm Method 6. Time-Varying Reliability Evaluation Based on the Tm Method Since concrete failure is regarded as f < f , the performance function is deduced as Equation (18), core c f ＜f core c wherSince e x is the concret random e failur variable e is reg of arded compr as essive strength. , the perf Accor orm ding ance tofthe uncti cumul on isative deduced as Eq density function uation (1 (CDF) 8), where of the xgamma is the ra distribution, ndom variable the of failur compressiv e probability e strengt (p ) h is. Ac expr cording t essed aso the cumulative density function (CDF) of the gamma distribution, the failure probability ( ) is expressed as G(x) = x 18 (18) G(x) = x − 18 Z (18) a(t)1 (t)x p (t) = x e dx (19) f f a(t)1 a (t ) −1 − λ (t ) x (t) G((t)) p (t ) = x e dx (19) −a (t ) λ ( t ) Γ (α (t )) where G(x) is the performance function; and x is the random variable of compressive strength. where One G (x of ) is the perform the most important ance function; steps to and calculat x is the e p r(atndom v ) is to analyze ariable of the compre integral ssive of the strength probability . density function (PDF), leading to the diculty in probability computation. In this paper, a reliability p (t ) One of the most important steps to calculate is to analyze the integral of the probability analysis based on the TM method was proposed with a simpler calculation. density function (PDF), leading to the difficulty in probability computation. In this paper, a reliability The calculation steps of the time-varying TM reliability index ( (t)) [25] are 3M analysis based on the TM method was proposed with a simpler calculation. β (t ) 3M The calculation steps of the time-varying TM reli 2 ability index ( ) [25] are (t) = (3 9 + (t) 6 (t) (t))/ (t) (20) 2M 3M 3G 3G 3G β (t） = (3 − 9 + α （t） − 6α (t )β (t ) ) / α (t ) (20) 3M 3G 3G 2M 3G (t) = (t)/ (t) (21) 2M G G β (t） = μ (t ) / σ (t ) (21) 2 M G G (t) = (t)/(t) 18 (22) μ (t ) = α (t ) / λ (t ) − 18 (22) (t) = (t)/(t) (23) σ (t ) = [α (t ) − λ (t ) × 18] / α (t ) (23) = 2/ (t) (24) 3G α = 2 / α (t ) (24) where is the SM reliability index; is 3the G mean value of G(x); is the standard deviation of 2M G G G(x); and is the skewness of G(x). 3G μ σ β G(x) 2 M G G where is the SM reliability index; is the mean value of ; is the standard deviation of Figures 15 and 16 show the time-varying failure probability, time-varying reliability index, and the corr G(xesponding ) results of MC simulation G(x)( f = 18.0 N/mm ). Other failure probabilities and reliability 3G ; and is the skewness of . indices of dierent f can be developed using the same method. In this paper, an MC simulation was Figures 15–16 show the time-varying failure probability, time-varying reliability index, and the used to verify the accuracy of reliability. From the MC simulation, the errors of the failure probability corresponding results of MC simulation ( f = 18.0 N/mm ). Other failure probabilities and reliability and the TM reliability index were within 3%, as shown in Table 3. Obviously, from Figure 16, (t) 3M indices of different can be developed using the same method. In this paper, an MC simulation was used to verify the accuracy of reliability. From the MC simulation, the errors of the failure probability Appl. Sci. 2020, 10, 815 13 of 20 β (t ) 3 M and the TM reliability index were within 3%, as shown in Table 3. Obviously, from Figure 16, p (t) β (t ) β (t ) 2 M 3M had a better agreement than because skewness was considered in . Thus, and β (t ) 3M in this paper can be used to evaluate the structural safety with high accuracy. Table 3. Part of failure probabilities and reliability indices from 36 to 40 years. Error Error p (%) Serving year β 3M f 3M ( ) ( ) 36 7.69 0.09% 1.416 0.67% Appl. Sci. 2020, 10, 815 12 of 18 37 9.17 0.46% 1.320 0.09% 38 10.75 1.03% 1.229 0.42% 39 12.43 0.19% 1.142 0.92% had a better agreement than (t) because skewness was considered in (t). Thus, p (t) and (t) 2M 3M f 3M 40 14.19 0.53% 1.060 1.43% in this paper can be used to evaluate the structural safety with high accuracy. Appl. Sci. 2020, 10, 815 14 of 20 Figure 15. Time-varying failure probability and MC simulation ( f = 18.0 N/mm ). Figure 15. Time-varying failure probability and MC simulation ( f = 18.0 N/mm ). Figure 16. Time-varying reliability index and MC simulation ( f = 18.0 N/mm ). c 2 Figure 16. Time-varying reliability index and MC simulation ( ). f = 18.0 N/mm Table 3. Part of failure probabilities and reliability indices from 36 to 40 years. 7. Concrete Guaranteed Rate and Concrete Failure Time Serving Year p (%) Error (p ) Error ( ) 3M 3M f f On the basis of the building code calibration, concrete quality is determined by the guaranteed 36 7.69 0.09% 1.416 0.67% rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed 37 9.17 0.46% 1.320 0.09% 38 10.75 1.03% 1.229 0.42% rate was applied in this paper. The concrete failure time can be obtained when the qualified rate ( ) 39 12.43 0.19% 1.142 0.92% p p g q 40 14.19 0.53% 1.060 1.43% is equal to the guaranteed rate ( ). As shown in Figure 17, is represented by the dimension of the shaded area. μ = 22 V = 0.09 Figure 17. Normal distribution of and . is obtained through the probability degree (d) [39], which reflects the relative degree of limit error, as shown in Table 4. Appl. Sci. 2020, 10, 815 14 of 20 Figure 16. Time-varying reliability index and MC simulation ( f = 18.0 N/mm ). Appl. Sci. 2020, 10, 815 13 of 18 7. Concrete Guaranteed Rate and Concrete Failure Time 7. Concrete Guaranteed Rate and Concrete Failure Time On the basis of the building code calibration, concrete quality is determined by the guaranteed rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed On the basis of the building code calibration, concrete quality is determined by the guaranteed rate of above 95% [40]. Therefore, to evaluate concrete failure time based on reliability, the guaranteed rate was applied in this paper. The concrete failure time can be obtained when the qualified rate ( ) rate was applied in this paper. The concrete failure time can be obtained when the qualiﬁed rate (p ) is p p g q is equal to the guaranteed rate ( ). As shown in Figure 17, is represented by the dimension of equal to the guaranteed rate (p ). As shown in Figure 17, p is represented by the dimension of the g q the shaded area. shaded area. Figure 17. Normal distribution ofμ ==22 22 and V = 0.09. V = 0.09 Figure 17. Normal distribution of and . p is obtained through the probability degree (d) [39], which reﬂects the relative degree of limit is obtained through the probability degree (d) [39], which reflects the relative degree of limit error, as shown in Table 4. error, as shown in Table 4. Table 4. Probability degree and guaranteed rate. p (%) p (%) d d g g 1.35 91.15 1.65 95.05 1.40 91.92 1.70 95.54 1.45 92.65 1.75 95.99 1.50 93.32 1.80 96.64 1.55 93.94 1.85 96.78 1.60 94.52 1.90 97.13 Other probability degrees can be calculated by the linear interpolation. p (t) and d(t) are expressed as p (t) = 1 p (t) (25) d(t) = ((t) f )/(t) (26) (t) = (t)/(t) (27) where is the standard deviation. Figure 18 shows the relationship between p and p . From Figure 18, the failure time of q g f = 18.0 N/mm was expected to be 35.8 years. When (shown in Figure 14) was applied in the c 2 reliability calculation, the concrete failure time of f = 21.0 N/mm was 38.8 years, as shown in Figure 19. Appl. Sci. 2020, 10, 815 15 of 20 p (t ) d (t ) and are expressed as p (t ) = 1 − p (t ) (25) q f d (t ) = (μ (t ) − f ) / σ (t ) (26) (27) σ（t） = α (t) / λ (t) where is the standard deviation. Table 4. Probability degree and guaranteed rate. p p d g d g (%) (%) 1.35 91.15 1.65 95.05 1.40 91.92 1.70 95.54 1.45 92.65 1.75 95.99 1.50 93.32 1.80 96.64 1.55 93.94 1.85 96.78 1.60 94.52 1.90 97.13 Other probability degrees can be calculated by the linear interpolation. p p q g Figure 18 shows the relationship between and . From Figure 18, the failure time of f = 18.0 N/mm was expected to be 35.8 years. When (shown in Figure 14) was applied in the reliability calculation, the concrete failure time of was 38.8 years, as shown in Figure f = 21.0 N/mm Appl. Sci. 2020, 10, 815 14 of 18 Appl. Sci. 2020, 10, 815 16 of 20 Figure 18. Time-varying qualiﬁed rate and guaranteed rate of f = 18.0 N/mm . Figure 18. Time-varying qualified rate and guaranteed rate of f = 18.0 N/mm . Figure 19. Time-varying qualified rate and guaranteed rate of . Figure 19. Time-varying qualiﬁed rate and guaranteed rate of ff = =21.0 21.0N N/mm /mm . The failure probability and reliability index are equivalent indices of reliability. According to The failure probability and reliability index are equivalent indices of reliability. According to the the concrete failure time (35.8 years) calculated by the failure probability, the limit reliability index of concrete failure time (35.8 years) calculated by the failure probability, the limit reliability index of f = 18.0 N/mm was observed as 1.466. Other limit reliability indices of dierent f values can be c c f = 18.0 N/mm was observed as 1.466. Other limit reliability indices of different values can be observed through the same method. Failure mode II is deﬁned as the reliability index less than the observed through the same method. Failure mode II is defined as the reliability index less than the limit reliability index. limit reliability index. 8. Prediction Model of Equivalent Design Compressive Strength 8. Prediction Model of Equivalent Design Compressive Strength To test the accuracy of predicted results (concrete failure time) calculated by time-varying To test the accuracy of predicted results (concrete failure time) calculated by time-varying reliability, this paper applies the prediction model of equivalent design compressive strength ( f ). ce 0 0 By the deﬁnition of f , f f implies that p < p , at which point the concrete is regarded as out of c q g ce ce ce reliability, this paper applies the prediction model of equivalent design compressive strength ( ). service. McIntyre [41] calculated the equivalent design compressive strength from a set of core tested ' ' p ＜p f f ≤ f q g ce c ce By the de data by the finfollowing ition of steps: , implies that , at which point the concrete is regarded as 2 2 V = V 0.002 (28) out of service. McIntyre [41] calculated the equivalent design compressive strength from a set of core situ core in tested data by the following steps: 2 2 V = V − 0.002 (28) in core situ V = f V (29) in in situ， 30 situ — — (30) f = [1.724 − 4.138(V )] f ×1.08 in insitu ,17.5 situ， 30 c ,v (31) f = Rs × f c,v c (32) f = 1.481 f − 11 .24 ce insitu ,17 .5 in situ core where and are the coefficients of variation of the situ compressive strength and the core in situ， 30 test strength, respectively; is the converting coefficient of variation when there are fewer c,v than 30 tests; is the multiplying factor (see [41]); is the mean value of core strength (with Appl. Sci. 2020, 10, 815 15 of 18 V = f V (29) situ,30 situ in in f = [1.724 4.138(V )] f 1.08 (30) insitu,17.5 situ,30 c,v in f = Rs f (31) c,v c f = 1.481 f 11.24 (32) ce insitu,17.5 Appl. Sci. 2020, 10, 815 17 of 20 where V and V are the coecients of variation of the situ compressive strength and the core situ core in test strength, respectively; V is the converting coecient of variation when there are fewer than situ,30 in insitu ,17.5 core coefficient of variation = ); is the mean value of situ compressive strength (with 30 tests; f is the multiplying factor (see [41]); f is the mean value of core strength (with coecient of c,v variation = V ); f is the mean value ce of situ compressive strength (with coecient of variation coefficient ofcore variat insitu ion = 1 ,17.57.5%); and is the equivalent design strength. = 17.5%); and f is the equivalent design strength. ce core The in this paper is expressed as Equation (33) based on Figure 8. The V in this paper is expressed as Equation (33) based on Figure 8. core V (t ) = 0.07 × exp(0.033t ) (33) core V (t) = 0.07 exp(0.033t) (33) core The relative equivalent design compressive strength is expressed as The relative equivalent design compressive strength is expressed as R = f / f (34) f ce c R = f / f . (34) f ce R ＜ 1 Failure mode III occurs when . Based on Equations (28)–(34), failure mode III was reached Failure mode III occurs when R < 1. Based on Equations (28)–(34), failure mode III was reached 2 2 at 36.7 years when 2, and at 38.3 years when 2 (see Figure 20). These f = 18.0 N/mm f = 21.0 N/mm at 36.7 years when f = c 18.0 N/mm , and at 38.3 years when f = c 21.0 N/mm (see Figure 20). These c c results were close to the calculated results of 35.8 years in Figure 18, and 38.8 years in Figure 19, results were close to the calculated results of 35.8 years in Figure 18, and 38.8 years in Figure 19, respectively. The calculated results were in good agreement because both the mean and coefficient respectively. The calculated results were in good agreement because both the mean and coecient of variation were considered in the two prediction models. Moreover, the failure modes of the two of variation were considered in the two prediction models. Moreover, the failure modes of the two models were based on the concrete guaranteed rate. models were based on the concrete guaranteed rate. 2 2 22 Figure 20. Relative equivalent design concrete strength ( f = 18.0 N/mm and f = 21.0 N/mm ). Figure 20. Relative equivalent design concrete strength (cf = 18.0 N/mm and cf = 21.0 N/mm ). c c 9. Example 9. Example f of beam I and beam II of dierent buildings in Yokohama, Japan, was 18.0N/mm , and the c 18.0N / mm service years of beam of beam Ι and bea I and m beam II of different II were 31 buildings and 38 years, in Yokoha respectively ma, Jap . an, was , and the service years of beam Ι and beam II were 31 and 38 years, respectively. In this example, f , t , and t were 18.0 N/mm , 31 years, and 38 years, respectively. Climate c 1 2 statistics were collected on a related weather website. f t t 1 2 18. 0 N/mm In this example, , , and were , 31 years, and 38 years, respectively. Climate statistics were collected on a related weather website. 9.1. Calculation Ι Using Equations (4) and (10), the relative compressive strengths of beam Ι and beam II were calculated as Rs = 1.36 Rs = 1.34 , . 2 ,Ⅰ 2 , Ⅱ According to failure mode Ι, both beams were still in service. Appl. Sci. 2020, 10, 815 16 of 18 9.1. Calculation I Using Equations (4) and (10), the relative compressive strengths of beam I and beam II were calculated as Rs = 1.36, Rs = 1.34. 2, 2, According to failure mode I, both beams were still in service. 9.2. Calculation II Using Equations (14) and (15) and (20)–(24), the corresponding reliability indices were calculated as = 1.9787, = 1.2288. 3M 3M , , According to failure mode II, beam I was in service, whereas beam II had been destroyed. 9.3. Calculation III Using Equations (4), (10), and (28)–(34), the corresponding equivalent design compressive strengths are calculated as R = 1.395, R = 0.9. f 2, f 2, According to failure mode III, the result of the safety evaluation coincided with that obtained in calculation II. On the basis of the analysis and discussion above, the result of Calculation I is dierent from others because only the mean value is considered. Therefore, the ﬁrst result is inconvincible. From the second and third results, beam I is relatively secure. However, beam II is suggested to be retroﬁtted. It is suggested that the member of failure probability of less than 0.5 is supposed to be replaced rather than strengthened. 10. Conclusions (1) The environmental factors (temperature and humidity) are considered in the prediction model of MVCS. (2) Compared with other distributions, such as a normal distribution and lognormal distribution, the gamma distribution is applied to describe the distribution of compressive strength with the highest R-square. (3) The time-varying TM reliability, with higher accuracy in contrast to SM method, is proposed to predict the concrete failure time. (4) Compared with the results of MC simulation, the prediction model of time-varying reliability based on the TM method is accurate, and the errors are within 3%. (5) Tested by the model for calculating an equivalent design compressive strength, the errors of the concrete failure time calculated by time-varying reliability are acceptable. 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Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength

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Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength

Prediction of Concrete Failure Time Based on Statistical Properties of Compressive Strength

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