## 4 Design Equations for Strengthening with ASFRM Overlay
### Design Equations for Diagonal Tension Strength  
In most masonry prisms strengthened with an ASFRM overlay for diagonal tension tests, cracks were developed only in the masonry layer, as shown in Figure 8c,d. This failure occurs because of the principal tensile stress in the diagonal direction, and the corresponding strength can be determined as the sum of the diagonal tension strength of the masonry layer and the sum of bond stress along a strip of unit width in the main diagonal of the interface between the masonry layer and ASFRM overlay, as illustrated in Figure 17, assuming that the horizontal shear stress in the masonry prism is negligible. Therefore, an equation for the diagonal tension strength comprising the two components is proposed as follows: First, the bond stress is assumed to exhibit a parabolic distribution, in which the stresses increase from zero at the corner to the peak at the center of the specimen. The peak bond stress \(\tau_{max}\) is assumed to be 1.5 times the average bond stress. Therefore, the diagonal tension strength for a strengthened masonry prism is defined as the sum of the diagonal tension strength of the bare masonry prism and the average interface bond stress exerted in the area of unit width, in accordance with Figure 17, as follows:  
\[S_{E,\rho}=\frac{n_{s}\frac{\sqrt{2}}{2}b(1)\frac{2}{3}\tau_{max}+S_{m}(1)t}{t (1)}=S_{m}+n_{s}0.71\frac{b}{t}\tau_{aver}, \tag{14}\]  
where \(S_{E,\rho}\) is the expected diagonal tension strength for the masonry prism strengthened with an ASFRM overlay; \(S_{m}\) is the diagonal tension strength for a bare masonry prism obtained from tests; \(b\) and \(t\) are the width and thickness of the bare masonry prism, respectively, as illustrated in Figure 17. In addition, \(n_{s}\) equals 1 or 2 for strengthening on one side or both sides, respectively. Finally, \(\tau_{aver}\) is the average bond strength, which can be expressed as \(2/3\tau_{\max}\) for the parabolic stress distribution, and can be obtained from the test, as summarized in Table 7.  
Figure 16: Predicted and test compressive strengths for prisms strengthened on both sides: (**a**) Red clay brick masonry; (**b**) concrete brick masonry. The diagonal tension strengths predicted using Equation (14) and the corresponding test strengths were compared, as shown in Figure 18, in which specimens strengthened with a 30-mm-thick ASFRM overlay on one side and both sides were included, and specimens with anchors for shear connectors were excluded. The trend lines for both the expected and lower-bound strengths based on Equations (4) and (5) are shown in Figure 18. The average ratios of the test strength over the nominal strength were 1.15 and 1.12 for the strengthened masonry prisms in red clay bricks and concrete bricks, respectively. Hence, Equation (14) underestimates the diagonal strength conservatively. The effect of filling the vertical joint partially when plastering the ASFRM might have contributed to the underestimation of the diagonal tension strength; this is because \(S_{m}\) in Equation (14) is based on the strength obtained from specimens with weak vertical mortar joints. The expected and lower-bound diagonal tension strengths are summarized in Tables 8 and 9, respectively. Equation (14) assumes that the ASFRM overlay does not fail after the peak load is reached. However, diagonal cracks were formed on the ASFRM overlay at the peak load in the case of concrete brick masonry strengthened on both sides. Therefore, the diagonal tension strength calculated using Equation (14) must be limited by an upper bound. Figure 17: Distribution of stress contributing to diagonal tension strength for strengthened prisms. Figure 18: Predicted and test diagonal tension strengths: (**a**) Red clay brick masonry; (**b**) concrete brick masonry.