## 4 Results and Discussions
### Ultimate Compressive Strength of Cracked Plates
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Crack Size**} & \multicolumn{1}{c}{**Intact**} & **10\%** & **25\%** & **50\%** \\ \hline \multirow{6}{*}{Longi. crack} & \multicolumn{2}{c}{Present} & 0.688 & 0.664 & 0.641 & 0.623 \\ \cline{2-6}  & \multicolumn{2}{c}{Equation (6a)} & 0.688 & 0.655 & 0.607 & 0.525 \\ \cline{2-6}  & \multicolumn{2}{c}{Difference} & 0.000 & 0.014 & 0.053 & 0.157 \\ \cline{2-6}  & \multicolumn{2}{c}{Present} & 0.688 & 0.686 & 0.625 & 0.532 \\ \cline{2-6}  & \multicolumn{2}{c}{Equation (5)} & 0.688 & 0.619 & 0.516 & 0.344 \\ \cline{2-6}  & \multicolumn{2}{c}{Equation (6b)} & 0.688 & 0.676 & 0.612 & 0.505 \\ \cline{2-6}  & \multicolumn{2}{c}{Difference with Equation (5)} & 0.000 & 0.098 & 0.174 & 0.353 \\ \cline{2-6}  & \multicolumn{2}{c}{Difference with Equation (6b)} & 0.000 & 0.015 & 0.021 & 0.051 \\ \hline \multirow{3}{*}{Trans. strength} & \multicolumn{2}{c}{Present} & 0.270 & 0.262 & 0.241 & 0.204 \\ \cline{2-6}  & \multicolumn{2}{c}{Equation (5)} & 0.270 & 0.243 & 0.203 & 0.135 \\ \cline{1-1} \cline{2-6}  & \multicolumn{2}{c}{Difference} & 0.000 & 0.073 & 0.158 & 0.338 \\ \hline \end{tabular} Difference: (1 – Present/equation).  
\end{table} 
Table 3: Comparison of ultimate strength (\(\sigma_{\text{u}}/\sigma_{\text{Y}}\)) of cracked plates under uniaxial compression with the existing formula (in Mpa). combined cracks led to a substantial reduction in both strength and stiffness, indicating their critical influence on structural integrity. Figure 6: Structural behavior of plate under longitudinal uniaxial compression. Figure 7: Structural behavior of plate under transverse uniaxial compression. Figure 7 shows the effect of cracks on the structural response under transverse uniaxial compression. Similar to the longitudinal strength, the transverse load capacity of the  
Figure 8: Effect of crack on ultimate strength of plate under biaxial compression. structure was affected by the cracks. Interestingly, the transverse cracks did not significantly influence the transverse response (Figure 7b) because, after buckling, most of the applied load was carried by the plate edges (Figure A10). The stress distribution confirms that under transverse loading, the plate edges primarily resist the applied load, regardless of crack size. As a result, cracks that form are situated in regions with inherently lower stress, minimizing their impact on the overall structural response. Thus, no significant difference was identified between the effects of the longitudinal and combined cracks whose length and breadth were smaller than 50% of the structural dimensions. Combined cracks sized 50% of the structural dimensions made a quarter of the plate separate from the main structure (Figure A15). When the load is applied along the crack direction, strength deterioration progresses more rapidly, as stress concentration around the crack increases, making propagation easier. Conversely, if the loading and crack directions differ, the crack may close, or the stress may be redistributed to other regions, slowing the reduction in strength. Also, it is observed that under combined loading, the structural load-carrying capacity diminishes more rapidly than in cases with a single-direction crack. This is because the interaction between the two crack directions disrupts the stress redistribution mechanism, leading to localized instability. Figure 8 and Tables A1-A3 present the results of biaxial loading on cracked plates, highlighting the influence of crack size and type on their ultimate compressive strength. As shown in Figure 8d, small cracks had a negligible effect on the ultimate strength under biaxial compressive loads, regardless of crack orientation. Among the different crack types, the combined crack had the most significant impact, followed by the longitudinal crack. In contrast, the effect of the transverse crack varied with the loading ratio (Figure 8b) and had no influence on the ultimate strength in the transverse direction.