## 3 Design Equation for the HPFRC Concrete Jacketing  
The repair strategy, which does not replace the removed transverse steel reinforcement; it restores the removed concrete parts with a thickness of about 100 mm in the entire plastic hinge region with an HPFRC jacket. This jacket does not modify the column dimension. The shear strength of the repaired RC components (V\({}_{\text{Rd}}\)) can be calculated by Equation (9).  
\[\text{V}_{\text{Rd}}=\text{V}_{\text{Rd,d}}+\text{V}_{\text{Rd,cc}} \tag{9}\]  
where V\({}_{\text{Rd,d}}\) is the contribution of the concrete jacket and V\({}_{\text{Rd,cc}}\) is the contribution of the original column core.  
The jacket contribution may be calculated by Equation (10) in CNR-204/2006 [55]:  
\[\text{V}_{\text{Rd,d}}=\left[\frac{0.18}{\gamma_{\text{C}}}\text{k}\cdot \left(100\cdot\rho_{\text{I}}\cdot(1+7.5.\frac{\text{f}_{\text{Fulk}}}{\text{f} _{\text{ck}}})\cdot\text{f}_{\text{ck}}\right)^{\frac{1}{3}}+0.15\cdot\sigma_{ \text{cp}}\right]\cdot\text{b}_{\text{w}}\cdot\text{d} \tag{10}\]  
where \(\gamma_{\text{c}}\) is the partial factor, \(\rho_{\text{I}}\) is the reinforcement ratio for the longitudinal reinforcement in tension, \(\text{f}_{\text{ck}}\) is the cylindrical characteristic compression strength of the concrete, \(\text{k}=1+\sqrt{(200/\text{d})}\) is the factor that considers the size effect, \(\sigma_{\text{cp}}\) is the average stress acting on the concrete cross section, \(\text{f}_{\text{ck}}\) is the characteristic tensile strength of the concrete matrix, \(\text{f}_{\text{Fulk}}\) is the characteristic value of the ultimate residual tensile strength, \(\text{b}_{\text{w}}\) is the beam width, and \(\text{d}\) is the distance from the extreme compression fiber to the centroid of tension steel reinforcement.  
The repaired columns jacket had a hallow section with a thickness (\(\text{t}_{\text{j}}\)) equal to 100 mm, which is the thickness of the external removed concrete parts. The product \(\text{b}_{\text{w}}\cdot\text{d}\) in Equation (10) should be substituted by the area \(\text{A}_{\text{e}}\) calculated by Equation (11), proposed by [PERSON] et al. [59] for hollow sections.  
\[\text{A}_{\text{e}}=\frac{\left(1+\left(\frac{\text{D}_{\text{i}}}{\text{D}} \right)^{2}\right)\cdot\left(1-\frac{\text{D}_{\text{i}}}{\text{D}}\right)}{ \left(1+\left(\frac{\text{D}_{\text{i}}}{\text{D}}\right)^{3}\right)}\text{ A}_{\text{n}} \tag{11}\]  
where D is the external diameter of the column, \(\text{D}_{\text{i}}\) is the inside diameter of the section and \(\text{A}_{\text{n}}\) is the net area of the concrete section.  
Alternatively, it can be assumed that \(\text{b}_{\text{w}}=2\cdot\text{t}_{\text{j}}\) and \(\text{d}=0.8\cdot\text{D}\) according the Canadian standards (CSA) [60] and the American standards (ACI design codes) [61; 62].  
Figure 6: (**a**) Tensile strength based on the bending test results; (**b**) load-CTOD curve example. Figures extracted from CNR DT 204/2006 [55] and from UNI 11039-2 (2003) [52].  
The shear strength contribution of the original specimen concrete core (V\({}_{\rm Rd,cc}\)) can be evaluated by Equation (12), presented by [PERSON] et al. [63]:  
\[{\rm V}_{\rm Rd,cc}=0.167\cdot\left(1+\frac{\rm N}{\left(14\cdot{\rm A}_{\rm g} \right)}\right)\cdot\sqrt{\rm f_{c}}\cdot{\rm b}_{\rm w}\cdot{\rm d} \tag{12}\]  
where b\({}_{\rm w}\cdot{\rm d}\) is assumed to be equal to 80% of the area of the column core, N is the axial load on column, A\({}_{\rm g}\) is the gross section of column, and r and r\({}_{\rm s}\) are the radii of the section and of the circular layer where the longitudinal steel rebar is located, respectively.