# Prediction of Deflection of Shear-Critical RC Beams Using Compatibility-Aided Truss Model
## 1 Introduction  
A lot of existing structures worldwide remain vulnerable to shear stress due to the lack of shear analysis and design technologies. Although the shear design technology itself is advanced, there are many cases in which the shear capacity of the structure is made insufficient by errors in structural design and construction or problems in management. In recent times, existing structures tend to be repaired and rehabilitated to save raw materials and reduce carbon emissions and energy consumption. The effectiveness of such repair and rehabilitation depends on an accurate evaluation of the stress and strain states of damaged structures. The deflection of flexure-critical RC beams has been widely investigated. [PERSON] [1] proposed a deflection analysis method that reflects the decrease in the stiffness of RC beams due to flexural cracks using the effective moment of inertia. Subsequently, many other researchers have conducted studies to revise [PERSON]'s formula [2, 3, 4, 5, 6, 7]. Among them, [PERSON] et al. [8, 9] reported that [PERSON]'s equation revealed a substantial deviation in the steel ratio amounting to 1% or less and proposed a new equation for the effective moment of inertia. In the ACI building code, [PERSON]'s equation had replaced [PERSON]'s equation in 2019 [10, 11]. Flexure-critical RC beams have been designed to exhibit sufficient ductility by flexural yielding of the tension reinforcement. However, in shear-critical RC beams with low shear strength, shear deterioration occurs before flexural yielding. Shear deterioration causes an increase in shear deformation that cannot be evaluated by using the elastic bending theory. Therefore, to analyze the deformation of shear-critical RC beams, a shear analytical model that can predict both the shear strength and shear deformation of RC beams is required. A theoretical approach to the shear analysis of RC members uses the 45-degree truss model proposed by [PERSON] and [PERSON] as a basis and continues to develop steadily. Unlike bending, shear is not easy to analyze because it is largely related to the tensile propertiesof concrete. Most shear analysis theories, including the 45-degree truss model, cannot predict shear deformation because they only consider the stress equilibrium condition. The first shear analytical model to predict the shear deformation of RC members was the compression field theory (CFT) developed by [PERSON] and [PERSON] [12] based on the tension field theory proposed by [PERSON]. The CFT can predict the shear stress and shear strain of RC elements by considering the strain compatibility conditions in addition to the stress equilibrium conditions. Subsequently, [PERSON] and [PERSON] [13] developed a modified compression field theory (MCFT) that reflects the tensile properties of concrete, significantly improving the accuracy of the shear analysis. A truss model that predicts shear stress and shear deformation by considering the strain compatibility conditions as well as the stress equilibrium conditions is called a 'compatibility-aided truss model'. This model can be divided into rotating and fixed angle truss models, depending on whether the crack angle is fixed. The rotating angle-softened truss model (RA-STM) developed by [PERSON] [14] is included in the rotating angle truss model together with the MCFT. The fixed angle-softening truss model (FA-STM) proposed by [PERSON] and [PERSON] [15] is the first compatibility-aided truss model considering a fixed crack angle, which assumes that the directions of principal compressive stress and fixed cracks are the same. The difference between directions of the principal compressive stress and fixed cracks increases as the load increases when the difference in the bidirectional steel ratio is generally substantial, such as in RC beams. [PERSON] et al. [16] proposed a transformation angle truss model (TATM) to solve this problem. The TATM can predict the shear stress and shear strain of RC beams at each load stage.