# Probabilistic Studies on the Shear Strength of Slender Steel Fiber Reinforced Concrete Structures
## 1 Introduction  
Over the years, the use of steel fiber reinforcements to improve the performance of reinforced concrete (RC) members in shear and tension has been extensively researched. Many studies have demonstrated that the inclusion of steel fiber reinforcements in concrete significantly enhances the tensile, flexural and shear capacity; improves the ductile and post-cracking behaviour and increases the energy absorption properties [1; 2; 3; 4; 5; 6]. The various investigations conducted in [7; 8; 9; 10; 11; 12] reported that the inclusion of fiber reinforcements in reinforced concrete beams significantly improves the shear capacity. An experimental study conducted by [PERSON] and [PERSON] [13] revealed that fiber reinforcements could potentially substitute the minimum stirrups required in conventional reinforced concrete members to ensure a ductile failure. The presence of steel fiber reinforcement in concrete enhances shear resistance by resisting and redistributing inclined tensile stresses along the diagonal cracks, improves post cracking resistance capacity and reduces the diagonal crack width and spacing. The reduction in the crack spacing suggests that the inclusion of steel fibers could result in a possible reduction or elimination of the size effect in beams without stirrups, where shear capacity decreases with increasing beam effective depth [8; 14; 15]. The efficiency of steel fiber reinforcements to increase shear capacity depends on several factors which include the fiber properties,fiber content and bond stress versus slip response of fibers [8]. Other parameters affecting the shear resistance of SFRC beams consist of the effective member depth \(d\), the concrete compressive strength \(f_{c}\), the aggregate size \(d_{ag}\), the shear-span to effective depth \(a/d\), and the longitudinal reinforcement ratio \(\rho\)[16]. The \(a/d\) has been shown to significantly influence the failure mode and mechanism of reinforced concrete structures [17]. Three patterns of shear failure--diagonal-tension, shear-tension and shear-compression--have generally been observed when steel fiber reinforcements are used in beams with or without stirrup reinforcement [8]. Despite the numerous studies and findings, the behaviour of SFRC beams in shear remains a complex phenomenon that requires further attention. Shear failure poses a crucial safety issue since it may develop swiftly with no prior signs of damage. Consequently, shear design provisions should be reliable and accurate. Numerous design expressions have been introduced in the current design standards [18; 19; 20] and literature [21]. However, developing a precise and acceptable design formulation for shear capacity of SFRC beams is still an evolving process. A summary of most of the proposed shear design models for SFRC beams without stirrups in current literature and design guidelines can be found in [21]. Majority of the existing shear design formulations are either empirical or semi-empirical based, except for the models based on the Modified Compression Field Theory (MCFT) [22; 23], the Dual Potential Capacity Model [24], plasticity-based models [25; 26] and the modified Multi-Action Shear Model [27]. The comparison of the various SFRC shear resistance models, as presented in [21; 27], indicate the difficulty and complexity of the accurate prediction of shear capacity of SFRC beams. Not only do the resistance predictions of the various procedures differ from each other, but they also vary from the experimentally observed shear strength. Given these disparities, adequate quantification and description of model uncertainties taking into account the differences between the predicted and the measured shear resistance is crucial, especially for reliability-based analysis. Structural engineering design is performed under a certain degree of inherent uncertainty. The influence of such uncertainties is quantified and accounted for using statistical and probabilistic approaches [28; 29].