# Nonlinear Finite Element Analysis Formulation for Shear in Reinforced Concrete Beams
## 1 Introduction  
The modern seismic design provisions for reinforced concrete (RC) structures enforce ductility requirements to meet target limit states, avoiding brittle failure [1]. As a typical shear failure of an RC member is often very brittle, more accurate descriptions of shear behavior are imperative in achieving proper safety goals. Therefore, the shear performance of RC members should be clearly investigated and adequately modeled. In general, the global responses of reinforced concrete (RC) structures are often obtained by means of a frame analysis model, where the members can be idealized by a one-dimensional beam-column element. The conventional method in considering beam shear utilizes the Timoshenko beam with a shear coefficient. Most shear coefficients known today are strain-driven functions, such as parabola shear strains [2; 3; 4; 5]. Such strain-driven shear functions do not properly describe nonlinear shear evolution. As a result, the conventional method rarely shows the shear strength degradation and shear failure that are often observed in loaded RC beams and columns. Many shear models have been proposed to improve such shear simulations, as follows. [PERSON] et al. [2; 3] developed a flexibility-based element to model shear behavior and its interaction with axial force and bending moment in RC beams and columns. Based on the fiber sectiondiscretization, the state of a section was determined by numerically integrating biaxial constitutive relations over the cross section. While the longitudinal strain field was determined from the plane section assumption, the shear distributions over the cross section were defined by strain-driven shear functions, such as constant shear strain, or parabolic shear strain functions. Despite the use of the micro-plane where the two-dimensional realization of shear was possible, the stain-driven shear functions gave little hint of shear degradation. A similar development was made by [PERSON] et al. [PERSON] et al. (2016) but based on the mixed finite element method using the Hu-Washizu variational form, and later extended by [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) for RC members. As in [PERSON]'s model [PERSON] (1999), the shear strain distribution over the cross section was assumed to be parabolic. The equilibrium-driven stress function was first introduced in the beam formulation by [PERSON] and [PERSON] [PERSON] and [PERSON] (2010) called the dual-section analysis. This method well predicted the shear stress distributions over the cross section of an RC beam. The shear flow of the cross section was determined from the normal stress difference between two cross sections. [PERSON] (1999) proposed the longitudinal stiffness method, where the shear flow or the normal force gradients were directly related to the sectional properties and the sectional shear force. It was a challenging task to write a consistent element formulation that utilized equilibrium-driven stress functions. The method worked best in the sectional analysis. However, both [PERSON] and [PERSON] (1999) and [PERSON] (1999) formulations have not yet been studied in the conventional finite element method. Recently, finite element analysis software programs [PERSON] (1999); [PERSON] and [PERSON] (2000); [PERSON] (2000) have been developed based on the finite element formulations proposed by the afore-mentioned researchers, and they were utilized by many researchers for their numerical simulations on shear behavior of RC members. [PERSON] et al. [PERSON] et al. (2010) have conducted finite element analyses to investigate the shear stress concentration phenomenon near a tapered cross section, and [PERSON] et al. [PERSON] et al. (2010) carried out a hybrid simulation testing on a precast concrete shear wall, which combines the structural test and numerical simulation. In addition, [PERSON] et al. [PERSON] et al. (2010) performed numerical simulations on masonry infilled RC buildings considering construction stages by using ABAQUS software.