## 4 Numerical Results  
In this section, we performed a large numerical investigation of the problem for an FG-FRCL beam subjected to a uniform temperature rise \(\Delta T=50.0\), and made of five layers with width \(b=0.40\) m, length \(L=4.0\) m, total thickness \(H=0.50\) m, thickness of each FGM layer, \(h_{f}=0.10\) m and thickness of the composite layer \(h_{c}=0.3\) m. The upper and lower layers were made of an FGM varying through the thickness from a pure ceramic (\(Si_{3}N_{4}\)) to a pure metal (SUS304) with a volume fraction exponent, \(p=0.40\). The middle layers were made of three glass fiber-reinforced composite layers with [90/0/90] lay-ups. Additionally, the maximum amplitude of the nonlinear oscillation read as \(A_{0}=0.001\) m. The material properties of the structure are presented in Tables 1 and 2, in line with Refs. [18; 43].  
In order to validate the results of the current work, a comparison study was carried out against the available literature. The linear and nonlinear fundamental frequencies were, thus, determined from Equation (63), for simply supported composite laminated beams (\(h_{FG}=0\)) and simply supported FGM beams (\(h_{c}=0\)), by neglecting the thermal load. The results were obtained systematically for different values of slenderness ratio, in line with predictions by [PERSON] and [PERSON] [34], [PERSON] et al. [44] and [PERSON] et al. [45], as reported in Tables 3 and 4, respectively. Based on the results in these two tables, it is worth observing  
lowest nonlinear natural frequency. The important point is that FG-FRCL beams are characterized by the highest nonlinear natural frequency compared with FGM beams and composite laminated beams, lighter than FGM beams and with a higher resistance against temperature rise (due to the presence of a ceramic phase) compared to composite laminated beams.  
Figures 3 and 4 show the effect of the power index and uniform temperature rise on the nonlinear fundamental frequency of simply supported FG-FRCL beams. As can be observed from the results, for an increased power index, the nonlinear fundamental frequency tended to increase, whose effect became more pronounced for lower slenderness ratios. At the same time, for an increased temperature, the fundamental frequency decreased, whose effect was more pronounced for increased slenderness ratios.  
Figures 5 and 6 represent the effect of the stacking sequence on the nonlinear natural frequency and thermal buckling temperature of simply supported FG-FRCL beams, respectively. The results demonstrate that FG-FRCL beams with [0/0/0] lay-ups exhibited the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, in sequence. A [90/90/90] lay-up, instead, provided the lowest nonlinear natural frequency and the lowest critical buckling tempera  
Figure 3: Effect of the power index on the nonlinear fundamental frequency.  
Figure 2: Variation of nonlinear natural frequency versus slenderness ratio.  
ture of the structure. Moreover, the results showed that the critical buckling temperature of the system increased significantly, for an increased power index.  
Figure 7 represents the phase trajectory of three different types of simply supported beams including the FGM beam (n = 1.0), FG-FRCL beam (n = 1.0, and [90/0/90]) and composite laminated beam ([90/0/90]). As can be seen, the phase trajectory exhibited a limit cycle, denoting that the system was free from dynamic chaos. The results show that the velocity of the nonlinear oscillation of the FG-FRCL beams assumed the highest value, followed by FGM beams and composite laminated beams, in sequence.  
Figure 4: Effect of the uniform temperature rise on the nonlinear fundamental frequency.  
Figure 5: Effect of stacking sequence on the nonlinear fundamental frequency.  
Figure 6: Variation of the critical buckling temperature versus power index.