Reliability analysis of shear strength equations of RC beams

Abstract

Keywords: Reinforced Concrete; Beam; Shear Strength; Probability; Reliability; Failure; Survival

Introduction

The reliability or serviceability may be assured only in terms of probability that the available capacity (R) will be adequate to withstand the lifetime maximum load (S). The aim of the reliability analysis is to ensure the event (R > S) during the lifetime of the engineering system [1-8]. The difference between R and S is included in the member design through safety criteria used in the structural codes. As this must be accomplished under conditions of uncertainty, the assurances of performance is realistically possible only in terms of probability. In general, therefore, probability analyses will be necessary in the development of such probability-based designs [2]. The uncertainties inherent in the constituent material strengths and densities, the member geometry, the applied loads, and the errors in load and strength calculations give rise to uncertainties in the resistance of a RC member as well as in the loads that act on

it. As a result, the nominal strength and loads computed by the designer differ from the actual ones. These safety criteria are provided either implicitly as those used for the working stress design format or explicitly as those for the ultimate strength design format. In this study, the reliabilities of the shear strength equation of RC beams were investigated by constructing the performance function between those equations and experimental results using a second-moment approach [9-10]. Numerical analyses were conducted for iterative solution with non-normal distribution. In practice, non-normal distributions are transformed into equivalent normal distributions. Conversely, according to international statistical data, it is assumed that the material strengths are log-normal and, the other variables are normal [2,6-8,11-13]. Also, it is assumed that the random variables are statistically independent, and the correlation effects are not taken into account.

*Corresponding author. *E-mail address: noyan@yildiz.edu.tr This paper was recommended for publication in revised form by Regional Editor Eyup Debik Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Failure And Survival Probabilities

The performance of the structural system is represented by a performance function 𝑍 = 𝑔(𝐗) = 𝑔(𝑋1, 𝑋2, … 𝑋𝑛), where 𝐗 = (𝑋1, 𝑋2, … 𝑋𝑛) is a vector of basic random variables. These basic variables may be loads, material strengths, dimensions, etc. which are taken into account in the structural system [14]. The limit state of system may be defined as 𝑍 = 0. 𝑍 > 0 and 𝑍 < 0 are respectively survival and failure states, respectively. The probabilities of the failure 𝑝𝐹 and the survival 𝑝𝑆 can be determined with integral of the joint probability density function 𝑓𝐗(𝐱), in spaces 𝑍 < 0 and 𝑍 > 0 [1,2,9,10]: (1)

(3) The calculation of the probabilities of survival or failure requires the knowledge of the joint probability distribution. In practice, this information is often unavailable or difficult to obtain for reasons of insufficient data. For this reason, the failure and survival probabilities can be determined by the second-moment approach whose name is derived from the definition of variance [1,2,9,10]. In second-moment approach used for predicting the probabilities of failure and survival, performance function Z is expanded in the Taylor series. Then performance function Z is linearized by taking into account only first-order terms in the Taylor’s series expansion. Thus, the mean 𝑚𝑍 and standard deviation 𝜎𝑍 of the linear function are determined. If the probability distributions of random variables are not normal, such distributions are transformed into equivalent normal distributions 𝑁(𝑚𝑍, 𝜎𝑍). For the limit state 𝑍 = 0, the value of the standard normal distribution variable is 𝑠 = (0 − 𝑚𝑍)/𝜎𝑍 = −𝑚𝑍/𝜎𝑍. The 𝛽 = 𝑚𝑍/𝜎𝑍, is called as reliability index. The probability of survival, therefore, becomes 𝑝𝑆 = Φ(𝛽) and the corresponding probability failure is 𝑝𝐹 = Φ(−𝛽) = 1 − Φ(𝛽). In order to predict the probabilities of survival or failure using second-moment approach, the reliability index is determined. The cumulative density function of the standard normal distribution is Φ(. ) [1,2,6-10]. In this study, it is assumed that the distributions of compressive strength of concrete (𝑓𝑐), the yield strength of stirrups (𝑓𝑦𝑤), and the shear strength of RC beam (𝑣𝑢) are log-normal. For this reason, the log-normal distribution is transformed into equivalent normal distribution. The mean value (𝜆𝑋) and standard deviation (𝜁𝑋) of the log-normal distribution can be determined by using Equations 4 and 5 as follows: (4)

(5) Then, the mean and standard deviation for the equivalent normal distribution of original variable can be expressed as [1,2]: (6) (7) In iterative second-moment approach, the point on the failure surface with a minimum distance to the origin of reduced variates is the most probable failure point. This distance equals to the reliability index 𝑑𝑚𝑖𝑛 = 𝛽 [15,16]. The performance function 𝑍 = 𝑔(𝐗) = 𝑔(𝑋1, 𝑋2, … 𝑋𝑛) is expanded in the Taylor series at a point 𝐱∗, which is on the failure surface 𝑔(𝐱∗) = 0, the partial derivatives are evaluated and only first-order terms are taken into at account. In result, the performance function Z becomes (8) The mean value and variance of the Z are obtained by using Equations 9 and 10. (9)

(10) The dimensionless sensitivity coefficients 𝛼𝑖 can be expressed as the ratio of the uncertainty 𝜎𝑋𝑖 in the random variable 𝑋𝑖 to the total uncertainty 𝜎𝑍 as follows: (11)

If the probability distribution of a random variable with mean 𝑚𝑋𝑖 , standard deviation 𝜎𝑋𝑖 and coefficient of sensitivity 𝛼𝑖 is normal, the value of standard normal distribution variable is 𝑠 = −𝛼𝑖𝛽 for a given reliability index 𝛽. Thus, the design value of the variable 𝑋𝑖𝑑 can be determined by using Equation 12. (12)

Properties of RC Beams For slender RC beams (𝑎/𝑑) ≥ 2.5, geometrical and material properties of RC beams with stirrups are shown in Table 1, where 𝑓𝑐 is the concrete compressive strength, 𝜌𝑤 is the stirrup ratio, 𝑓𝑦𝑤 is yield strength of stirrup reinforcement, 𝜌 is longitudinal reinforcement ratio, 𝑎/𝑑 is the shear span-to-effective depth ratio, 𝑑 is the effective depth, 𝑏𝑤 is the web width, and 𝑣𝑢 is the test shear strength. All the beams failed in shear. Moreover, the frequency distributions of the selected variables are illustrated in Figure 1.

Sigma J Eng Nat Sci, Vol. 42, No. 1, pp. 141−152, February, 2024

Sigma J Eng Nat Sci, Vol. 42, No. 1, pp. 141−152, February, 2024

Sigma J Eng Nat Sci, Vol. 42, No. 1, pp. 141−152, February, 2024

Shear Strength Models In the design of RC frame elements, it is necessary to prevent shear failure mechanisms. The question of what mechanisms of shear transfer will contribute most to the resistance of a particular beam is difficult to answer [25]. Most of the shear design equations provide a simple superposition of stirrup and concrete strength [26-30]. The following procedure outlines the guidelines recommended by ASCE-ACI 426 [31] in order to determine the shear strength of RC beams. The governing equation from ACI318 [26] states that the shear capacity must exceed the shear demand as shown in Equation 13. (13) The nominal shear strength is derived from two components: concrete and stirrups. This relationship is given as follows: (14) in which 𝑣𝑐 is the shear strength of concrete; and 𝑣𝑠 is the shear strength of stirrup based on yield, respectively. The shear strength of stirrup, 𝑣𝑠, in case of vertical stirrups can be derived from basic equilibrium considerations on a 45˚ truss model with constant stirrup spacing and effective depth [25]. Summary of different shear strength models for RC slender beams is shown in Table 2.

As shown in Table 2, in ACI318 [26], TS500 [27] and ENV 1992 [29], (𝑎/𝑑) was not taken into account in predicting of the shear strength of the beam, unlike the equation proposed by Zsutty [30]. The size effect as a variable is considered only in the EN 1992:2004 [28] equation. Contrary to the ACI318 [26], TS500 [27] and Zsutty [30] equations, the ENV 1992 [29] and EN 1992:2004 [28] equations have been predicted by reducing the contribution of stirrups to the shear strength. Figure 2 compares the predicted shear strength with the experimental shear strength values obtained from tests. The experimental and predicted shear strengths for existing test data yield large scatter in the results, especially for increasing shear strength of the beams. The six different code requirements and researchers’ predictions are compared with the test results of 49 beams with stirrups. Probabilities of Failure and Survival with SecondMoment Approach The survival or failure probabilities of the equations determining the shear strength of RC beams were calculated by the second-moment approach described in the previous section. In this study, 𝑅 and 𝑆 are the experimental load-carrying capacity and the estimate of the load carrying capacity of the beam, respectively. For this reason, the statistical evaluation has been conducted by considering the beams satisfying, 𝑅 − 𝑆 > 0.

Table 2. Summary of different shear strength models for RC slender beams References

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Figure 2. Predicted shear strength values versus test shear strength values. In this reliability assessment, the equation resulting in the largest difference between 𝑅 and 𝑆 will be the most reliable one. In other words, the equation resulting the smallest failure probability will be the most reliable one. In the probabilistic evaluation, primarily, the performance function 𝑍 = 𝑅 − 𝑆 > 0 has been established. Therefore, the basic random variables included in the performance function are 𝑓𝑐, 𝑓𝑦𝑤, 𝜌𝑤, 𝜌, 𝑑, 𝑎/𝑑 and 𝑣𝑢, which are used in the shear strength equations proposed by ACI318 [26], TS500 [27], EN 1992:2004 [28], EN 1992 [29], ENV1992 [29] and Zsutty [30]. Uncertainties of Random Variables In general, the variations in the properties of RC beams depend on the construction quality control and environmental conditions. Both shear strengths obtained through experiments and equations were modeled as random variables to perform a probability-based analysis. In modeling those parameters as random variables, the values of standard deviations were determined based on the studies available in the literature and codes, and are summarized in Table 3. The standard deviation of concrete compressive strength, 𝜎𝑓𝑐 , under average construction quality control usually depends on the 𝑓𝑐 and varies in between 0.10𝑓𝑐 and 0.21𝑓𝑐 through the literature. According to TS500 [32], 𝜎𝑓𝑐 ranges from 3.1 MPa to 6.25 MPa depending on the 𝑓𝑐. ACI318 [26] recommends to increase the value of 𝜎𝑓𝑐 in case that the number of samples is less than 30, and proposes 5.2MPa for 𝑓𝑐 < 21 MPa, 6.34MPa for 21MPa ≤ 𝑓𝑐 ≤ 35 MPa and a value larger than 6.34 MPa based on the 𝑓𝑐. The 𝜎𝑓𝑐 is taken as 0.10𝑓𝑐 by Nowak and Szerszen [33] and Ribeiro and Diniz [34], 0.11𝑓𝑐 by Hao et al. [35], 0.12𝑓𝑐 by Neves et al. [36], 0.13𝑓𝑐 by Val et al. [37], 0.15 fc by Mirza [38], Mirza et al. [39], Mirza and MacGregor [40], Mirza and MacGregor [41], 0.16𝑓𝑐 by Val and Chernin [42], 0.20𝑓𝑐 by Melchers [43] and 0.21𝑓𝑐 by Ellingwood [44]. It is taken as

0.10𝑓𝑐 and 0.15𝑓𝑐 for cases 1 and 2, respectively, in the present study to model variations of 𝑓𝑐. The standard deviation of reinforcement yield strength 𝜎𝑓𝑦 ranges from 0.05𝑓𝑦 to 0.15𝑓𝑦. The 𝜎𝑓𝑦 are taken as 0.05𝑓𝑦 by JCSS [45], 0.08𝑓𝑦 by Val et al. [37] and Low and Hao [46], 0.08𝑓𝑦 − 0.11𝑓𝑦 by Ostlund [47], MacGregor et al. [48] and 0.15𝑓𝑦 by Mirza [38], Mirza et al. [39], Mirza and MacGregor [40], Mirza and MacGregor [41]. It is taken as 0.10𝑓𝑦𝑤 in the present study to model variations of 𝜎𝑓𝑦𝑤. Although the reinforcement ratios depend on the structural dimensions, in this study they are assumed to be statistically independent from each other and from the other random structural parameters. In Hao et al. [35] study, it is assumed that standard deviation of longitudinal reinforcement ratio, 𝜎𝜌, is 0.10𝜌 and standard deviation of stirrup ratio, 𝜎𝜌𝑤, is 0.15𝜎𝜌𝑤. In the present study, 𝜎𝜌 and 𝜎𝜌𝑤 are taken as 0.10𝜌 and 0.15𝜎𝜌𝑤 , respectively. In Enright and Frangopol [49] studies, it is assumed that the standard deviation of effective depth, 𝜎𝑑, is 0.03𝑑.

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[17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [18] [18] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [20] [21] [21] [21] [21] [22] [22] [22] [22] [23] [23] [23] [23] [24] [24] Mean Minimum Maximum

A-1 A-2 B-1 B-2 C-1 C-2 CRB-1 1WCRA-1 1WCRB-1 1WCRC-1 1WCA-1 1WCB-1 1WCC-1 2WCA-1 29a-2 29f-2 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R24 R25 R28 C305-D0 E2l E3l E4l E5l C3 R3 J3 Y3 B50-7-3 B150-3-3 B100-7-3 B150-7-3 ST6 ST18

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[17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [17] [18] [18] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [19] [20] [21] [21] [21] [21] [22] [22] [22] [22] [23] [23] [23] [23] [24] [24] Mean Minimum Maximum

A-1 A-2 B-1 B-2 C-1 C-2 CRB-1 1WCRA-1 1WCRB-1 1WCRC-1 1WCA-1 1WCB-1 1WCC-1 2WCA-1 29a-2 29f-2 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R24 R25 R28 C305-D0 E2l E3l E4l E5l C3 R3 J3 Y3 B50-7-3 B150-3-3 B100-7-3 B150-7-3 ST6 ST18

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Figure 3. Reliability index 𝛽 of the predicted equations for cases 1 and 2 . Table 6. Reliability rankings according to the probability of failure (𝑝𝐹) Ranking

The standard deviations of structural dimensions are taken as 0.03𝑑 by Low and Hao [46] and Hao et al. [35], 0.03𝑑 by Ribeiro and Diniz [34]. In the present study, 𝜎𝑑 and 𝜎𝑎/𝑑 are taken as 0.03𝑑 and 0.03(𝑎/d), respectively. In Hognestad [50] and Mirza [38] studies, it is assumed that the standard deviation of strength, 𝜎𝑣𝑢, due to test procedure is 0.04𝑣𝑢, this value is used in this study..

Evaluation of Failure Probability By taking into account the standard deviations and mean values of the variables, the performance function Z is separately formed for each shear strength equation [2630]. Then, the mean value 𝑚𝑍 and standard deviation 𝜎𝑍 of the performance function Z are determined. Thus, the reliability index 𝛽 is a function of the ratio 𝑚𝑍/𝜎𝑍. The probability of survival, therefore, becomes 𝑝𝑆 = Φ(𝛽) and the corresponding probability of failure is 𝑝𝐹 = 1 − Φ(𝛽). As shown in Tables 4-5, numerical analysis was carried out for non-normal solution with iteration. Log-normal distributions are transformed into equivalent normal distributions as explained in the previous section. Mean, minimum and maximum reliability indexes 𝛽 of the predicted equations are shown Figure 3 for cases 1 and 2, respectively. Mean, minimum and maximum probabilities of failures, obtained for each beam were evaluated and sorted from minimum to maximum, as shown in Table 6. As can

be clearly seen in Table 6, EN 1992:2004 equation yields the lowest probability of failure and Zsutty equation has the highest probability of failure. For case 1 𝜎𝑓𝑐 = 0.10𝑓𝑐, the targets β of RC beams for ultimate states are calculated as 4.10 (range 1.81-7.11) according to ACI318, 2.22 (range 0.14-5.29) according to TS500, 8.84 (range 5.19-12.92) according to EN 1992:2004 and ENV1992, 4.87 (range 2.05-7.67) according to EN 1992,

Conclusion

Considering the limited data collected from literature, the following conclusions can be drawn from the results of this study: • It can be said that EN 1992:2004 equation yields the lowest probability of failure and Zsutty equation has

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the highest probability of failure based on the reliability rankings provided by non-normal iterative solution. By comparing the reliabilities of shear strength equations of RC beams for two different cases(𝜎𝑓𝑐 = 0.10𝑓𝑐 and 𝜎𝑓𝑐 = 0.15𝑓𝑐), there were no significant changes in the reliability rankings. For case 1 (𝜎𝑓𝑐 = 0.10𝑓𝑐), the mean target values of 𝛽 for ultimate states are calculated as 4.10 according to ACI318, 2.22 according to TS500, 8.84 according to EN 1992:2004 and ENV1992, 4.87 according to EN 1992,

Nomenclature

Ratio of the shear span to effective depth of beam The web width (mm) The effective depth (mm) Compressive strength of concrete (MPa) Probability density function, Yield strength of longitudinal reinforcement (MPa) Yield strength of stirrup reinforcement (MPa) Performance function Mean Mean value of equivalent normal distribution Failure probability Survival probability Capacity Demand The standard normal variable Coefficient of variation The shear strength of concrete Cracking shear strength of beam Nominal shear strength The shear strength of stirrup Random variable Reduced variates Most probable failure point

Greek symbols 𝛼 Sensitivity coefficient 𝛽 Reliability index 𝜙 Strength reduction factor 𝜆𝑅, 𝜆𝑆 Capacity, demand. Mean value of lognormal distribution 𝜌 Longitudinal reinforcement ratio (%) 𝜌𝑤 Stirrup ratio (%) 𝜎 Standard deviation 𝜎𝑅, 𝜎𝑆 Capacity, demand. Standard deviation of normal distribution 𝜎𝑁𝑋𝑖 Standard deviation of equivalent normal distribution 𝜁𝑅, 𝜁𝑆 Capacity, demand. Standard deviation of lognormal distribution Subscripts 𝑐 Refers to concrete cr Refers to crack 𝑛 Refers to nominal 𝑠 Refers to steel 𝑤 Refers to web 𝑦 Refers to yield

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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