## 3 Numerical Results
### Thin-Walled Single-Cell FG Box Beam  
In this section, the mechanical behaviour of a cantilever thin-walled single-cell FG beam subjected to large deflection under an axial load is investigated. The geometric properties of the FG beam are as in Figure 1, and the cross section of the FG beam consisting of 5 layers is shown in Figure 9. The outer layer of the beam is ceramic, while the inner layer is metal. The properties of c and m materials are detailed above. The volume fraction of the c material of the beam follows the power law as [55]\[BV_{c}(x,z)=\left(1-\frac{x,z}{t}\right)^{m} \tag{21}\]  
for the volume fraction of the m material of the beam,  
\[V_{m}(x,z)=1-V_{c}(x,z) \tag{22}\]  
where \(m\) is the compositional gradient exponent in the \(x\)- and \(z\)-directions, and \(t\) is the beam thickness. The Mori-Tanaka scheme [56] is used for grading the material properties according to this volume fraction. Details are not included in the name of brevity. Large Deflection of Cantilever Thin-Walled Single-Cell FG Box Beam for Post-Buckling  
In this section, two different compositional gradient exponent values of the FG beam are taken, m = 0.5 and 5, representing the ceramic-rich and metal-rich composition, respectively. For these compositional gradient exponents, the displacement and stress distributions of the FG beam subjected to the axial load effect are investigated in the NL regime. Figure 10 shows the equilibrium curves of the FG box beam with polynomial degrees L4 and L9 subjected to axial loading effect for different compositional gradient exponents. The behaviour characteristics of the displacement component at L4 and L9 are similar for both compositions. However, P* levels increase approximately 2.5 times in ceramic-rich composition compared to metal-rich composition. Figure 10: The equilibrium curves for L4 and L9 of single-cell FG beam subjected to axial loading, P* = 4\(PL^{2}/\pi^{2}EI\). Figure 9: Thin-walled FG beam cross section. Figure 11 shows the distribution of axial and shear stress components for two different P* values of L4 and L9 of ceramic-rich FG box beam subjected to axial loading for the NL regime. The variation of the axial stress component is similar for both loads at L4 and L9. The distribution of the shear stress component along the \(z\)-axis at different load effects is similar for L4, but for L9, as the load increases, a sudden rise occurs in the lower parts of the beam. Figure 12 shows the distribution of axial and shear stress components for two different P* values of L4 and L9 of the metal-rich FG box beam subjected to axial loading for the NL regime. The variation of the axial stress component along the \(z\)-axis is similar for L4 and L9, but as the load increases, the level of the stress component decreases at the upper parts of the beam, and the distribution changes. The distribution of the shear stress component is similar for both loads in L4 and L9. Figure 13 shows the distribution of the axial and shear stress components in L and NL regimes for different P* values of the ceramic-rich FG box beam in L9. The distributions of the axial stress components for \(P_{6}^{*}\) are close to each other for L and NL regimes. For \(P_{18}^{*}\) and \(P_{34}^{*}\), the distribution of the axial stress component is similar along the \(z\)-axis in the L regime. In the NL regime, axial stress along the \(z\)-axis changes the behaviour of the beam towards the top at \(P_{18}^{*}\) and \(P_{34}^{*}\) loads. As seen in Figure 13, when the shear stress is examined, the distribution along the \(z\)-axis is similar for all load values in L behaviour. In the NL regime, while \(P_{6}^{*}\) and \(P_{18}^{*}\) are similar, at \(P_{34}^{*}\) the levels decrease at the bottom of the beam. The distribution of axial and shear stress components in L and NL regimes for different P* values of the metal-rich FG box beam is shown in Figure 14 for L9. The distribution of the axial stress component in L and NL regimes is similar at \(P_{6}^{*}\).