# Static Strength and Buckling Analysis of an Aircraft Support
## 5 Buckling Analysis  
The aircraft will be overloaded during take-off or landing [12], especially in the case of an emergency landing of large fuselage weight, which will undoubtedly bring a huge test to the load-bearing components of the aircraft. Through the buckling analysis, the structural instability of the bearing assembly under large load can be further obtained. In this paper, the buckling analysis is used to study the critical load of the main bearing area of the bearing assembly [13], as well as the instability of the bearing area when the critical load is applied. Buckling analysis includes linear eigenvalue buckling [14] analysis and nonlinear buckling analysis [15]. The linear eigenvalue buckling analysis does not consider the influence of geometric nonlinearity on the equilibrium equation and geometric equation [16], and it  
Figure 8: The strain curve under the test load. Figure 9: S6 comparing the experimental values with the numerical analysis results. solves the buckling mode and critical load of linear instability. The critical load can be used as the reference value of the nonlinear buckling analysis load [17]. Figure 10 shows the buckling analysis curve. The critical load point A is the linear eigenvalue buckling value, and the first peak point B of the solid curve is the nonlinear buckling value. The nonlinear buckling value is lower than the linear buckling value, which is generally about 40-60% of the linear buckling value. After reaching the first nonlinear buckling value, the structure will enter the instability region. Compared with linear buckling, nonlinear buckling is more accurate and in line with the actual situation. In this paper, the eigenvalue method is used to analyze the linear buckling of the aircraft bearing assembly. The eigenvalue of the first-order buckling mode is taken as the critical load, which is used as the reference value of the nonlinear buckling analysis load. In the linear buckling analysis, it is assumed that the material of the bearing assembly remains in an elastic state, and the relationship of eigenvalue buckling is as follow:  
\[[\{K_{e}\}+\{\rho\}(\{K_{u}\}+\{K_{G}\})]\{\phi\}=0 \tag{3}\]  
where \(K_{e}\) is the elastic stiffness matrix of the structure, \(K_{u}\) is the initial displacement matrix, \(K_{G}\) is the initial stress matrix, \(\rho\) is the critical load, and \(\phi\) is the displacement characteristic vector. Ignoring \(K_{u}\), the characteristic equation of buckling stability analysis is obtained:  
\[(\{K_{e}\}+\{\rho\}\{K_{G}\})\{\phi\}=0 \tag{4}\]  
Finally, considering the eigenvalues \(\rho_{1},\rho_{2},\rho_{3},\rho_{4},\rho_{5},\rho_{6}\dots\dots\dots\), the eigenvalue \(\rho_{1}\) is the first-order critical load of linear buckling analysis, and the corresponding displacement mode \(\phi_{1}\) is the first-order buckling mode. In this paper, \(\rho_{1}\) is used as the load reference value of nonlinear buckling analysis. In the numerical analysis, the Riks arc length method is used to analyze the nonlinear buckling of the bearing assembly, and the critical load of nonlinear buckling is solved. The iterative equation is structured as follows:  
\[\Delta U^{i}K_{t}=\lambda_{t+\Delta t}P-N_{t+\Delta t}^{i-1} \tag{5}\]  
where \(\Delta U^{i}\) is the iteration increment of current displacement, \(K_{t}\) is the tangent stiffness matrix of the structure at time t, \(\lambda_{t+\Delta t}\) is \(t+\Delta t\) load scale factor at time \(t\), \(P\) is external load vector, and \(N_{t+\Delta t}^{i-1}\) is \(t+\Delta t\) Internal force vector of structural node at time \(t\). To solve \(\lambda_{t+\Delta t}\), additional constraint conditions need to be added, and the constraint equation is obtained by constraining the load level and displacement vector. The displacement increment \(\Delta U^{i}\) is  
Figure 10: Buckling analysis curve. A is the linear eigenvalue buckling value, B is the nonlinear buckling value. combined with the load scale factor \(\lambda_{t+\Delta t}\).