## 4 Design Equations for Strengthening with ASFRM Overlay
### Design Equations for Compressive Strength
#### 4.1.2 Strengthening on Both Sides  
The compressive strength of masonry prisms strengthened on both sides can be calculated based on Equation (6) and experimental observations. In tests for the compressive strength, masonry prisms strengthened on both sides experienced compressive failure in the masonry layer and cracks at the masonry-to-overlay interface in the strength degradation region beginning at the strength cap in the load-displacement curves. Cracks or other damages were not observed on the ASFRM overlay. Based on the experimental observations, it was assumed that the full compressive strength of the ASFRM overlay was not developed, and that only a fraction of the strength corresponding to the ultimate strain of the masonry layer contributed to the total compressive strength. Therefore, the total compressive strength is expressed as follows, assuming that the ASFRM remains elastic:  
\[\sigma_{E,\rho}=\sigma_{m}+E_{ASFRM}\varepsilon_{mu}\frac{A_{ASFRM}}{A_{m}}, \tag{13}\]  
where \(E_{ASFRM}\) is the elastic modulus of the ASFRM (as summarized in Table 3); \(\varepsilon_{mu}\) is the ultimate strain of the masonry layer, for which 0.0035 and 0.0025 were applied to the red clay bricks and concrete bricks, respectively, in accordance with TMS 402 [28].  
The compressive strengths predicted using Equation (13) and the corresponding test strengths were compared, as shown in Figure 16, in which the trend lines for both the expected and lower-bound strengths based on Equations (4) and (5) are shown. The ratios of the test strength over the nominal strength were 0.83 and 1.04 for the strengthened masonry prisms in red clay bricks and concrete bricks, respectively. In other words, Equation (13) overestimates the compressive strength by approximately 17% after the red clay brick masonry was strengthened; therefore, it needs to be reduced by a factor of 0.83 before more test data are acquired. The overestimation is attributable to the assumption of elastic behavior in the ASFRM masonry. Nonetheless, the accuracy of the predicted compressive strength is expected to be improved by determining the elastic modulus corresponding to \(\varepsilon_{mu}\) in a more sophisticated manner, e.g., using the secant modulus at the critical strain. For strengthening on both sides, the expected and lower-bound compressive strengths with respect to the nominal strength are summarized in Tables 8 and 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type of Test**} & \multirow{2}{*}{\begin{tabular}{c} **Nominal** \\ **Strength** \\ **(Q\({}_{E,\rho}\))** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Expected** \\ **Strength** \\ **(Q\({}_{E}\))** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Lower-Bound** \\ **Strength** \\ **(Q\({}_{L}\))** \\ \end{tabular} } \\ \cline{2-4} Compressive & One side & Equation (12) & \(1.17\mathcal{Q}_{E,\rho}\) & \(1.0Q_{E,\rho}\) \\ \cline{2-4}  & Two sides & Equation (13) & \(1.04\mathcal{Q}_{E,\rho}\) & \(0.94\mathcal{Q}_{E,\rho}\) \\ \hline \multicolumn{4}{c}{Diagonal tension 1} & Equation (14) & \(1.12\mathcal{Q}_{E,\rho}\) & \(0.86\mathcal{Q}_{E,\rho}\) \\ \hline \hline \end{tabular}  
* Diagonal tension strength shall be limited by upper bound expressed in Equation (15).  
\end{table}
Table 9: Nominal strength and modification factors for concrete brick masonry.