## 4 Computational and Mathematical Aspects
### Constitutive Laws of Masonry Materials  
For the determination of the masonry compressive and tensile strength, several semi-empirical expressions are available in the literature. In the majority of these expressions, global effects contributing to the system resistance, such as buckling-effects or local-compression resistance are not considered. Recent studies [72,73] have shown that masonry codes underestimate the compressive strength of masonry with high coefficients of variation. For the estimation of the masonry strengths of low-strength masonry, with a single leaf, the following formula have been proposed by [PERSON] and [PERSON] [71]:  
\[f_{wc}=\xi\Big{[}\left(\frac{2}{3}\sqrt{f_{bc}}-\alpha\right)+\beta f_{mc} \Big{]}\ \left[\text{in MPa}\right] \tag{1}\]  
\[f_{wt}=\frac{2}{3}f_{mt} \tag{2}\]  
where  
\(f_{wc},f_{wt}\) are the compressive and tensile strength of masonry respectively;  
\(f_{mc},f_{mt}\) are the compressive and tensile strength of mortar, respectively;  
\(f_{bc}\) is the compressive strength of the block/stone or brick material;  
\(\alpha\) is a reduction factor due to non-orthogonality of blocks (\(\alpha=0.5\) for block stones & \(\alpha=2.5\) for rubble stones);  
\(\beta\) is a mortar-to-block factor (\(\beta=0.5\) for rough stones and \(\beta=0.1\) for very smooth-surface stones); and  
\(\xi\) is a factor expressing the adverse effect of thick mortar joints, \(\xi=1/[1+3.5(k-k_{0})]\) (\(k\) = volume of mortar/volume of masonry and \(k_{0}=0.3\)). However, for well-built and regular masonry structures, [PERSON] [74] proposed another expression for the estimation of the compressive strength:  
\[f_{wc}=[f_{mc}+0.4(f_{bc}-f_{mc})]\big{(}1-0.8\sqrt[3]{\alpha}\big{)}\ \text{for}\ f_{bc}>f_{mc} \tag{3}\]  
and  
\[f_{wc}=f_{bc}\big{(}1-0.8\sqrt[3]{\alpha}\big{)}\ \text{for}\ f_{bc}\leq f_{mc} \tag{4}\]  
where  
\(f_{bc},f_{mc}\) are the compressive strength of blocks and mortar, respectively; and  
\(\alpha=t_{jm}/h_{bm}\) is the ratio between average bed (horizontal) joint thickness \(t_{jm}\), and average block height \(h_{bm}\). Based on a large database of compressive test results on masonry prisms from the literature and using regression analysis or soft computing techniques such as Artificial Neural Networks (ANNs) and Fuzzy Logic (FL), a plethora of mathematical models have been proposed for the estimation of masonry compressive strength. Detailed state-of-the-art reports can be found in the works of [PERSON] and [PERSON] [72], [PERSON] and [PERSON] [73], and [PERSON] et al. [75]. A common feature of these proposals (Table 1) is that they only consider the compressive strength of brick and mortar. Recently, [PERSON] and [PERSON] [73] proposed a formula taking into account the majority of parameters affecting the masonry compressive strength. Namely, the authors, based on regression analysis on many test data (232 datasets) corresponding to the masonry unit strength of 3.1-127.0 MPa, mortar strength of 0.3-52.6 MPa and \(h\)/\(t\) ratio of 1.15-5.75, have proposed the following formula:  
\[f_{wc}=\frac{0.54\times f_{bc}^{1.06}\times f_{mc}^{0.004}\times VF_{b}^{3.3} \times VR_{mH}^{0.6}}{\left(\frac{h}{t}\right)^{0.28}} \tag{5}\]  
where \(VF_{b}\) is the volume fraction of brick and \(VR_{Mh}\) the volume ratio of bed joint to mortar defined by the following equations:  
\[VF_{b}=\frac{V_{u}}{V_{p}} \tag{6}\]  
where \(V_{u}\) is the volume of masonry units and \(V_{p}\) is the volume of prism. \[VR_{mH}=\frac{V_{mH}}{V_{mH}+V_{mV}} \tag{7}\]  
where \(V_{mH}\) is the volume fraction of mortar in horizontal joints and \(V_{mV}\) is the volume fraction of mortar in vertical joints. The volume fraction is obtained by dividing the respective volume with the corresponding total volume in the prism. The above proposed analytical formula (Equation (5)) seems