## 4 Calculation Method of Shear Carrying Capacity of UHPFRC Beam-Column Joint
### Components of Shear Carrying Capacity  
#### 4.2.1 Shear Force Carried by UHPFRC in Joint Core  
The shear carrying capacity of the UHPFRC in the joint core is dependent on the compressive strength of the UHPFRC diagonal strut, the value of which is the component force of the compressive  
Figure 8: Calculation method of shear carrying capacity of UHPFRC beam-column joint: (**a**) external actions of joint, (**b**) UHPFRC, (**c**) steel fiber, and (**d**) stirrup.  
strength of the UHPFRC diagonal strut in the horizontal direction. The inclination angle is \(\theta\); that is, the angle between the diagonal compressive strut and horizontal direction is defined as  
\[\theta=\arctan(\frac{h_{b}^{\prime\prime}}{h_{c}^{\prime\prime}}) \tag{4}\]  
in which \(h_{b}^{\prime\prime}\) and \(h_{c}^{\prime\prime}\) are the distances between the compression and tensional longitudinal reinforcement centerline of zone in the beam and column, as illustrated in Figure 8a. However, for exterior joints, \(h_{c}^{\prime\prime}\) has a different definition, namely, the distance from the extreme longitudinal reinforcement axis in the column to the bar extension axis at the free end of the 90-degree hooked bar.  
The effective area of diagonal strut \(A_{str}\) can be computed using Equation (5); that is,  
\[A_{str}=b_{s}\times a_{s} \tag{5}\]  
In the equation, \(b_{s}\) is the diagonal strut width and can be approximately regarded as the width of joint section \(b_{j}\), which is the lesser value of \((b_{b}+b_{c})/2\), \((b_{b}+0.5h_{c})\), and \(b_{c}\), in which \(b_{b}\) and \(b_{c}\) are the section widths of the beam and column, respectively, and \(h_{c}\) is the height of the column section. Moreover, \(a_{s}\) can be evaluated according to the equation \(a_{s}=\sqrt{a_{b}^{2}+a_{c}^{2}}\), in which \(a_{b}\) and \(a_{c}\) are the compression zone heights in the beam and column, respectively. If the beam adjacent to the joint core does not continue to burden the compression force owing to the appearance of a conspicuous bending plastic hinge and severe UHPFRC spalling when the joint fails, \(a_{b}\) is 0; if not, it can be considered as 1/5 of the beam section height in the calculation of \(a_{s}\). The determination of \(a_{c}\) adopts the equation proposed by [PERSON] and [PERSON] [6], namely, \(a_{c}=(0.25+0.85N)/f_{c}^{\prime}b_{c}h_{c}\), in which \(N\) is the axial compressive load acting on the upper column, \(f_{c}^{\prime}\) is the cubic compressive strength of UHPFRC, and \(b_{c}\) and \(h_{c}\) are the geometric dimensioning of the column section.  
Because of the complicated tensile-compressive cyclic stress state, the diagonal compressive strength and corresponding compressive strain of the concrete in the diagonally cracked joint core panel are less than those of the concrete in the uniaxial compressive test; that is, the strength in the principal compressive direction is softened by the principal tension in the perpendicular direction [26], and UHPFRC exhibits characteristics of decreasing. This study adopts the softening coefficient \(\zeta\) to account for this phenomenon. Existing research has demonstrated that \(\zeta\) is associated with \(f_{c}^{\prime}\) and the principal tension strain \(\varepsilon_{r}\). Based on the experimental results, the softening coefficient \(\zeta\) can be approximated by Equation (6), as follows [27]:  
\[\zeta=\frac{3.35}{\sqrt{f_{c}^{\prime}}} \tag{6}\]  
The shear carrying capacity of UHPFRC can be calculated using Equation (7):  
\[V_{c}=\zeta f_{c}^{\prime}A_{str}\cos\theta=3.35\sqrt{f_{c}^{\prime}}A_{str} \cos\left[\arctan(\frac{h_{b}^{\prime\prime}}{h_{c}^{\prime\prime}})\right] \tag{7}\]