## 4 Results and Discussions  
### Ultimate Compressive Strength of Cracked Plates  
[PERSON] et al. [9] and [PERSON] and [PERSON] [39] presented an approximate empirical formula to calculate the ultimate strength of cracked structures as expressed in Equation (5). \[\sigma_{u}=\frac{A_{intact}-A_{c}}{A_{intact}}\sigma_{u\_intact} \tag{5}\]  
where \(\sigma_{u\_intact}\) and \(\sigma_{u}\) are the ultimate strength of the intact and cracked structures, and \(A_{intact}\) and \(A_{c}\) are the cross-section area of intact structure and crack, respectively. The concept of the formula is to determine strength reduction by considering the ratio between an effective and intact cross-section area. This approach is applicable when the crack is oriented perpendicular to the applied load. [PERSON] et al. [29] proposed an empirical formula for the ultimate strength of one-directionally cracked plates under longitudinal compressions as Equations (6a) and (6b) considering directions and locations of crack. \[\sigma_{u}=[1-a_{1}]\cdot\left[1-a_{3}\Big{(}\frac{s}{b}\Big{)}^{2}\right] \cdot\sigma_{u\_intact}\;\;\text{for longitudinal crack}, \tag{6a}\] \[\sigma_{u}=[1-a_{2}]\cdot\left[1-a_{4}\Big{(}\frac{s}{b}\Big{)}^{2}\right] \cdot\sigma_{u\_intact}+0.045\;\;\text{for transverse and oblique crack}, \tag{6b}\]  
where \(\sigma_{u\_intact}\) and \(\sigma_{u}\) are the ultimate strength of the intact and cracked structures, \(a_{1}\) and \(a_{2}\) are coefficients related to the plate slenderness ratio and direction of crack, \(a_{3}\) and \(a_{4}\) are coefficients for the location of crack, \(s\) is the location of crack center in breadth direction, and \(b\) is the plate breadth. Figure 5 and Table 3 compare the ultimate strength of plates with perpendicular cracks under uniaxial compressions, as obtained by Equations (5), (6a) and (6b) and the results of this study. It shows that Equation (5) underestimates the ultimate strength, because it assumes a uniform stress distribution across the cross-section and a linear strength reduction based solely on crack size. However, in practical scenarios, stress redistribution is not perfectly uniform, and localized stress concentrations around the crack can cause  
Figure 4: Examples of applied initial deflection and distortion. nonlinear strength degradation rather than a simple linear reduction. This observation indicates the necessity of revising the equation to account for the effects of nonuniformly distributed stress across the cross-section. Otherwise, the results from Equations (6a) and (6b) and this study showed good agreement for small cracks, as Equations (6a) and (6b) was derived from FE simulations, which account for the structural response more accurately than simplified analytical approaches. Figures 6-8 present the effect of the crack type and size on the structural behavior of the plates. The ultimate compressive strength of the intact and cracked plates under uniaxial and biaxial compressive loads is summarized and compared. As shown in Figures 6 and 7a, cracks reduced both the longitudinal strength and stiffness of the plates, regardless of crack size and type. The impact of transverse and combined cracks on the ultimate compressive strength in the longitudinal direction was more pronounced than that of longitudinal cracks, as shown in Figure 5. However, cracks smaller than 10% of the plate dimensions had a negligible effect, as they did not significantly alter the stress distribution, as illustrated in Figures A1-A10 (Appendix A). In contrast, larger  
Figure 5: Effect of crack on ultimate strength of plate under uniaxial compression.