Optimized shear design equation for slender concrete beams reinforced with FRP bars and stirrups using Genetic Algorithm and reliability analysis
Introduction
The corrosion of steel shortens the life span of a reinforced concrete (RC) structure and repairing and/or rehabilitation of such structures becomes a costly undertaking. It is estimated that the repairing cost of all corroded steel reinforced structures in Canada would total $74 billion [1]. Over the last 20 years [2], the search for more durable, corrosion-resistant, and hence more cost-effective materials has driven research into Fiber Reinforced Polymer (FRP) reinforcement, due to its relatively high corrosion resistant properties compared to steel.
A high level of confidence is needed for shear design of FRP rebar, since unlike steel which yields, shear failure of brittle FRP can be catastrophic in comparison. The current guidelines and codes for FRP RC structures are designed to have a high level of confidence for beams not failing in shear. In other words, the current code equation provides over-conservative results in predicting shear and thus leads to higher reinforcement ratio in design causing reinforcement congestion problems and also making the overall construction expensive for FRP RC elements.
For calculating shear strength both American Concrete Institute (ACI) and Canadian Standards Association (CSA) codes assume that the concrete shear strength, Vc, and stirrup shear strength, Vfv, should be calculated independently and summed [3], [4]. The concrete shear equations are generally semi-empirical in nature. The equations contain different shear mechanisms that contribute to concrete shear strength and have coefficients for different material properties determined through regression analysis [5], [6]. Shear mechanisms known to affect the concrete shear strength of FRP RC beams include: shear friction, shear resistance of uncracked compression zone concrete, dowel action, arching action, and residual tensile stresses across cracks. Among all these mechanisms, the dowel action is generally not considered in the shear design equations for FRP. Since FRP is an anisotropic material the transverse direction dowel action for FRP is much less than that of steel and is ignored in the concrete shear strength equations [7]. On the contrary, the stirrup shear strength, Vfv, is generally determined with a parallel truss model [8].
Numerous models were reported in the literature to predict the shear behavior of slender beams reinforced with FRP. Hoult et al. [9] developed a shear model for FRP reinforced slender beams based on the modified compression field theory (MCFT) [10]. MCFT originally illustrates general shear behavior of cracked RC beams. Hoult et al. [9] found that the shear behavior of FRP reinforced slender beams without stirrups is similar to steel reinforced slender beams. This was proven later experimentally by Bentz et al. [11] and they found that the shear strength-strain effects on both FRP-reinforced and steel-reinforced members are similar. Perera et al. [12] proposed a new shear design equation using artificial intelligence techniques for RC beams strengthened externally by FRP U-wrap. Perera et al. [12] modified the existing ACI code shear equation [13] by Genetic Algorithm, using the strut and tie model as a basis of shear prediction. Machial et al. [14] performed a comparative study among the code equations and found them highly conservative, which might result in uneconomical use of FRP rebar in reinforced concrete structures. Nehdi et al. [5] proposed shear equations for beams with FRP as longitudinal reinforcement only. However, these proposed equations were based on limited available test data of 68 beams and overestimated the experimental shear strength compared to the new test data [14]. In addition, the reliability of those equations was unknown.
The present study proposes new coefficients for the shear design equations of ACI 440.1R-06 [13], CSA S806-02 [15] and CSA S6-09 [16] codes and tooth model by Reineck [17] by using Genetic Algorithm (GA) to improve their performance and accuracy for predicting shear strength of slender concrete beams reinforced with longitudinal and transverse FRP bars. In addition, resistance factors were recommended for the optimized shear equations from reliability analysis for the design purposes which ensure structural safety. Since FRP rebar is a relatively new material, guidelines for predicting shear strength of FRP reinforced concrete beams are often over-conservative [14] and thus increase the construction cost; this hinders the use of FRP. Additionally, many of the code equations were developed using limited experimental data. Currently more experimental data is available which can be utilized to improve the code equations. This study is critical as it will compare the performance between the existing code equations and the newly proposed equations based on the current larger database. This can lead to efficient and optimized design equations for slender FRP RC beams. However, it should be noted that the proposed shear equations do not attempt to model any physical reality of actual resisting shear mechanisms in FRP reinforced slender beams, which is far complex in nature. Genetic Algorithm optimization is well-suited to complex, multivariable systems and could be a useful tool for this purpose as it can predict better equation with high accuracy.
Section snippets
Current guidelines
Current shear design equations in the codes are semi-empirical in nature. The core of the equations are determined analytically and the coefficients are determined through regression analysis [5]. In the present paper, the code equations and models considered for optimization include the ACI 440.1R-06 [13], CSA-S806-02 [15], CSA-S6-09 [16], and Reineck’s tooth model [17]. These equations are listed in Appendix A.
Database
From a thorough literature review [2], [18], [19], [20], [21], [22], [23], [24],
Genetic Algorithms (GA)
This study utilizes Genetic Algorithm (GA) for optimization of shear equations. The GA optimization technique emulates biological evolution of survival of the fittest using three major processes: selection, crossover and mutation [38]. This is done using a population of possible solutions. The first process is to select the fittest individuals in the population for reproduction (crossover). In conjunction to crossover is elite selection where the specimens with the best fitness are passed on to
Results and discussion
The performances of the developed equations were compared to their original counterparts (Table 4, Table 5, Table 6) using the validation database. For the graphical comparisons the complete database was utilized to determine the overall performance using the χ value, which is the inverse of the slope of the Vcalc vs Vtest least squares linear regression line.
Reliability analysis
Reliability is a measure of the likelihood of a failure. In reliability model, load (Q) and resistance (R) are the random variables which can be determined from probability density functions. Nowak and Collins [48] described various methods to calculate the reliability, but in general reliability can be expressed by a limit state function g, such as Eq. (35),where g is the safety margin. If R and Q are normally distributed, then the reliability index (βI) can be expressed as,
Conclusion
This study proposes optimized shear equations for ACI, CSA S806-02, CSA S6-09 and Reineck tooth model, developed using GA, for FRP reinforced RC slender beams with and without transverse reinforcement. For simplicity, the optimized equations were kept similar to the corresponding original equations, which did not require additional calculations. With the development of the optimized equation, they were calibrated for the resistance factors in order to offer appropriate confidence levels in
Acknowledgments
Financial supports from National Science and Engineering Research Council (NSERC) through Undergraduate Student Research Award (USRA) and Discovery grant programs were critical to conduct this research, and are gratefully acknowledged. The authors would like to thank Ms. Alexandra Cheng for thoroughly reviewing the manuscript for grammar and punctuation.
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