## 2 AASHTO Seismic Shear Design Provisions  
The 2020 AASHTO LRFD Bridge Design Specifications [1] give procedures for the seismic design of reinforced concrete bridge columns. These procedures include a capacity design approach for determining the required shear strength, procedures for determining the amount of shear reinforcement, and the amount of confinement reinforcement in the plastic hinge region to ensure ductile response. The design provisions for determining the shear capacity are discussed and the predictions using these methods are compared with experimental results on shear-critical columns subjected to reversed-cyclic loading. The experimental results include columns with various cross-sectional shapes (circular, octagonal, and rectangular), reinforcement details, material properties, and axial load levels. In the 2020 AASHTO Specifications, the nominal shear resistance, \(V_{n}\), has a concrete contribution, \(V_{c}\), and a shear reinforcement contribution, \(V_{s}\). The equations for determining the nominal shear resistance of reinforced concrete columns are:  
\[V_{n}=V_{c}+V_{s}\leq 0.25f_{c}^{\prime}b_{v}d_{v} \tag{1}\]  
\[\text{For }\frac{P}{A_{g}f_{c}^{\prime}}\geq 0.10:\quad V_{c}=\beta\sqrt{f_{c}^ {\prime}}\ b_{v}d_{v} \tag{2a}\]  
\[\text{For }\frac{P}{A_{g}f_{c}^{\prime}}<0.10:\quad V_{c}=\beta\sqrt{f_{c}^ {\prime}}\ b_{v}d_{v}\bigg{(}10\frac{P}{A_{g}f_{c}^{\prime}}\bigg{)} \tag{2b}\]  
\[V_{s}=f_{y}A_{v}\frac{d_{v}\cot\theta}{s} \tag{3}\]  
where the factor \(\beta\) accounts for the shear resistance of the diagonally cracked concrete; \(b_{v}\) is the effective web width; \(d_{v}\) is the effective shear depth taken as the greater of \(0.72h\) or \(0.9d\), where \(d\) is taken as the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement in the tension half of the member; \(s\) is the spacing of transverse shear reinforcement; \(A_{v}\) is the area of transverse reinforcement within distance \(s\); and \(\theta\) is the angle of principal compressive stress with respect to the longitudinal axis of the member. It is noted that in the end regions of columns, Equation (2a) applies when the compressive axial load is greater than or equal to \(0.10f_{c}^{\prime}A_{g}\). However, for compressive axial loads less than \(0.10f_{c}^{\prime}A_{g}\), \(V_{c}\) decreases linearly from the value given in Equation (2a) to zero when the compressive axial load is zero (see Equation (2b)). This reduction factor accounts for the influence of reversed-cyclic loading in reducing the tensile stresses in the concrete without the beneficial effects of significant compressive axial load. The Specifications provide two different procedures for shear design. The Simplified Procedure assumes a value for the factor, \(\beta\), of 0.167 MPa. It also assumes an angle of principal compression, \(\theta\), of 45\({}^{\circ}\) in accordance with the traditional 45\({}^{\circ}\) truss model. The General Procedure is based on the Modified Compression Field Theory [2; 3; 4] and is adapted from the General Method of the Canadian Highway Bridge Design Code (CHBDC) [5]. To determine \(\beta\) and \(\theta\), the longitudinal strain, \(\varepsilon_{s}\), at the centroid of the tension reinforcement must first be determined. The equations for determining \(\beta\) and \(\theta\) for a reinforced concrete column, containing at least minimum amounts of transverse reinforcement, may be written as the following in SI units:  
\[\varepsilon_{s}=\frac{\frac{M_{u}}{d_{v}}+V_{u}+0.5N_{u}}{E_{s}A_{s}} \tag{4}\]  
\[\beta=\frac{0.4}{1+750\varepsilon_{s}} \tag{5}\]  
\[\theta=29+3500\varepsilon_{s} \tag{6}\]  
where the moment, \(M_{u}\) is taken as a positive value; \(N_{u}\) is the axial load (positive for tension and negative for compression); \(E_{s}\) is the elastic modulus of the longitudinal reinforcingsteel; and \(A_{s}\) is the area of the longitudinal reinforcing bars on the tension half of the member.