## 3 Numerical Results
### Thin-Walled Two-Cell FG Box Beam  
The geometric properties of the two-cell beam are as in Figure 15, and the cross section of the FG beam consisting of three layers is shown in Figure 20. Here, the outer layer of the beam represents the ceramic, the inner layer represents the metal and the middle layer represents the graded region. In this study, the buckling behaviour of the FG beam subjected to axial loading for two compositional gradient exponents is investigated. Equations (21) and (22) are used in the FG region grading and the ceramic and metal material properties are as given above.  
Large Deflection of Thin-Walled Two-Cell FG Box Beam for Post-Buckling  
In this section, the buckling behaviour of the cantilever twin-cell FG beam with large deflections subjected to axial loading is investigated for different compositional gradient exponents. The equilibrium curve of two-cell box FG beam with polynomial degree L9 is shown in Figure 21 for both NL and L regimes. The curves are plotted along the beam's \(y\)-axis at the position \(x=0\) and \(z=0\). Here, the displacement distribution for both compositional gradient exponents is similar.  
Figure 22 shows the distribution of axial and shear stress components for two different P* values in L9 of the ceramic-rich FG box beam subjected to axial loading for the NL regime. As seen in Figure 22a,b, although the distribution character is similar, when the load increases, the levels increase at the upper parts of the beam. The distribution of the shear stress component along the z-axis at different load effects is similar. Figure 23 shows the distribution of axial and shear stress components for two different P* values at L9 of the  
Figure 19: Distribution of axial and shear stress components for \(P^{*}=-0.126\), \(P^{*}=-0.155\), and \(P^{*}=-0.169\) values in L9 extensions at \(x=-a_{2}/2\) and \(y=L/2\) in the the two-cell composite beam subjected to axial loading; \(\sigma_{yy}^{*}=\sigma_{yy}/\sigma_{yy_{max}}\) and \(\sigma_{yz}^{*}=\sigma_{yz}/\sigma_{yz_{max}}\).  
Figure 20: Thin-walled two-cell FG beam cross section.  
metal-rich FG box beam subjected to axial loading. The distribution character of the axial stress component along the z-axis is similar for different loads, but the levels of the stress component increase in the upper parts of the beam as the load increases. The distribution of the shear stress component is similar for both loads.  
Figure 24 shows the distributions of the axial and shear stress components of the ceramic-rich FG box beam in L and NL regimes for different P* values in L9. The distributions of the axial stress components are similar for all loads in the L regime. The distribution of the axial stress component in the NL regime is similar for all loads. However, the load P* effects at lower levels. The shear stress distribution does not change for different load values both in the L regime and in the NL regime.  
Figure 21: The equilibrium curves for L9 of two-cell FG beam subjected to axial loading, P* = 4\(PL^{2}/\pi^{2}EI\).  
Figure 22: Distribution of axial and shear stress components for (**a**) P* = \(-\)0.0804 and (**b**) P* = \(-\)0.1446 loads in NL regime at L4 and L9 extensions at \(x=-a_{2}/2\) and \(y=L/2\) in the two-cell FG beam subject to axial loading; \(\sigma_{yy}^{*}=\sigma_{yy}/\sigma_{yy_{min}}\) and \(\sigma_{yz}^{*}=\sigma_{yz}/\sigma_{yz_{min}}\).  
Figure 25 shows the distributions of the axial and shear stress components of the metal-rich FG box beam in L and NL regimes for different \(P^{*}\) values in L9. The distributions of axial stress components are similar for all loads in both L and NL regimes. The shear stress distribution does not change for different load values both in the L regime and in the NL regime. The levels of the axial and shear stress components change depending on the change of the composition (Figures 24 and 25).