# Deep Embedment (DE) FRP Shear Strengthening of Concrete Bridge Slabs under Loads Close to Supports
## 6 Loading-Carrying Capacity Prediction Method  
In this study, a design model is proposed to predict the loading-carrying capacity of concrete slabs shear strengthened with deep embedment FRP bars. It was found in the test that the failure mode of concrete slabs could be varied from brittle shear failure to ductile flexural failure by using the deep embedment strengthening method. As a result, a two-way prediction approach, including bending and shear ultimate strengths, is adopted in this paper as shown below:  
**Flexural Capacity:**  
The flexural capacity of concrete slabs is predicted by using the equations in a bridge structure design code named BS5400 [35], which is given by:  
\[M_{b}=\rho fybd^{2}\big{(}1-0.59\rho f_{y}/f_{cu}\big{)} \tag{13}\]  
\[P_{b}=\frac{l_{0}}{(l_{0}-a)a}M_{b} \tag{14}\]  
Figure 21: Finite element (FE) crack patterns of test slabs. In Equation (13), \(\rho\) is the steel reinforcement percentage, \(b\) and \(d\) are width and effective depth of concrete slabs respectively, \(f_{y}\) and \(f_{cu}\) are yield strength of steel bars and concrete strength, respectively. In order to relate the bending moment (\(M_{b}\)) to the applied load (\(P_{b}\)), relevant elastic analysis is adopted as shown in Equation (14). **Shear Capacity:**  
The shear capacity of concrete slabs shear strengthened with deep embedment FRP bars is considered as the combination of the contribution of concrete and deep embedment FRP bars. In this study, the contribution of concrete to shear capacity is predicted by the design method in BS 5400 [35], due to the consideration of the influence of longitudinal steel reinforcement in this theoretical model, which is given by Equation (15) [35]. \[P_{cs}=0.79\cdot(100\rho)^{1/3}\cdot\left(\frac{400}{d}\right)^{1/4}\cdot\left( \frac{f_{cu}}{25}\right)^{1/3}bd \tag{15}\]  
A theoretical model proposed by [PERSON] et al. [20] is adopted to predict the shear capacity contributed by the application of deep embedment FRP shear strengthening method, as shown in Equation (16). \[\mathrm{P}_{FRPs}=k_{l}\cdot k_{s}\frac{A_{frp}\cdot E_{frp}\cdot\varepsilon_{ frp}\cdot D_{frp}(\sin\theta+\cos\theta)}{s_{frp}} \tag{16}\]  
where \(A_{frp}\) is deep embedment FRP bar cross-sectional area, \(d_{frp}\) is effective shear depth (the greater of 0.72 h and 0.9 d), \(\theta\) is inclination angle of FRP bars (90 degrees for this test), \(s_{frp}\) is spacing between FRP bars. \(\varepsilon_{frp}\) is effective strain in FRP bars, which can be calculated as below:  
\[\varepsilon_{frp}=\sqrt{\frac{8}{D_{frp}\cdot E_{frp}}\left(\frac{\tau_{m} \cdot s_{m}}{1+\alpha}\right)}\ \leq 0.004 \tag{17}\]  
In Equation (16), \(D_{frp}\) is diameter of FRP bars used in deep embedment strengthening, \(E_{frp}\) is elastic modulus of FRP bars. \(\tau_{m}\), \(s_{m}\), and \(\alpha\) are coefficients in the BPE (Eligehausen, Popov, and Bertero Model) modified bond-slip model [36] using in this theoretical model. The values of those coefficients are shown in Table 4 based on the bonding test results. In addition, \(k_{l}\) is a decreasing coefficient (\(0\leq k_{l}\leq 1\)) that represents the effect of FRP bars having an anchorage length shorter than the minimum anchorage length needed (\(L_{\mathit{eff}}\) as shown in Equations (19) and (20)). The effective anchorage length coefficient (\(k_{l}\)) can be determined using the following equations:  
\[k_{l}=\left\{ \begin{array}{cc}1&\frac{d_{\mathit{eff}}}{2}\geq L_{\mathit{ eff}}\\ &\frac{d_{\mathit{frp}}}{\sqrt{\frac{\varepsilon_{frp}\cdot D_{frp}}{2}}\frac{ \tau_{m}\cdot 1+\alpha}{\frac{1}{(1-\alpha)^{2}}}}&\frac{d_{\mathit{frp}}}{2} \mathrm{L}_{\mathit{eff}}\end{array} \right. \tag{18}\]  
\[L_{\mathit{eff}}(s_{m})=\frac{f(s_{m})\cdot D_{\mathit{frp}}}{4\tau_{m}}\cdot \frac{1+\alpha}{1-\alpha} \tag{19}\]  
\[f(s_{m})=\sqrt{\frac{8E_{frp}}{D_{frp}}\cdot\frac{\tau_{m}\cdot s_{m}}{1+\alpha}} \tag{20}\]  
Additionally, \(k_{s}\) accounts for the effect of the internal stirrup on the effective strain of strengthening FRP bars.