*3.2. Local Displacement and Stress Field*

According to Equation (2), the local displacement distributions within the unit cell at the geometric center of the 2D-RPM can be recovered as shown in Figure 8. It can be observed that the maximum and minimum values of *U*<sup>1</sup> and *U*<sup>2</sup> were located on opposite sides of the stiffener, and the overall displacement presented a centrosymmetric distribution trend. The smallest value of *U*<sup>3</sup> was located at the center of the stiffener, while the maximum value of *U*<sup>3</sup> was on the skin. The displacement distribution was centrally symmetric about the intersection point of the stiffeners. The maximum value of *U*<sup>3</sup> within the unit cell was 10.27 mm, and the error was about 1% compared with that of 3D-FEM presented in Section 3.1 (*U*<sup>3</sup> = 10.17 mm), indicating that the recovered displacement distribution is accurate and can be used to evaluate the location of maximum local displacement.

**Figure 8.** Local displacement fields within the unit cell at the geometric center of the plate (unit: mm).

Figure 9 shows the local stress fields within the unit cell at the geometric center of the plate recovered from Equation (8) and the 3D Hooke's law. It can be observed that the stiffeners played an important role in the process of load transfer, and there was a large stress concentration at the intersection of the stiffeners and the skin, showing a significant skin-stiffener effect. The stress distribution on the skin was relatively uniform, and there was no evident mutation. It was concluded that the stiffeners improved the bearing capacity of the panel, and the OSFP had a low weight and high strength compared to the ordinary panel.

Figure 10 shows the local von Mises stress and displacement distribution along Path 1 of the skin within the unit cell (as shown in Figures 8c and 9a) predicted by 2D-RPM and 3D-FEM. It can be observed that the local stress and displacement curves predicted by 2D-RPM and 3D-FEM agreed well, and the maximum error was less than 5%. The local stress at the junctions between the skin and the stiffeners decreased significantly, indicating that these regions were very incidental to be damaged.

**Figure 9.** Local stress fields within the unit cell at the geometric center of the plate (unit: MPa).

**Figure 10.** Comparison of von Mises stress and *U*<sup>3</sup> distribution along Path 1 within the unit cell at the geometric center of the OSFP.

#### *3.3. Free-Vibration Analysis*

Table 1 shows the the first four vibration modes and natural frequencies of the OSFP under the CCCC boundary condition predicted by 2D-RPM and 3D-FEM. The vibration modes of the 3D-FEM and 2D-RPM were in good agreement. For example, there were one and two half-waves along the *x*<sup>1</sup> direction, two half-waves along the *x*<sup>2</sup> direction, and two half-waves along the *x*<sup>1</sup> and *x*<sup>2</sup> directions for the first, second, third, and forth mode shapes, respectively, for both the 2D-RPM and 3D-FEM results. The natural frequencies of the 3D-FEM and 2D-RPM were also highly consistent, and the maximum error of the natural frequency was less than 6.83%. It was worth noting that the first-order vibration frequency showed relative big error compared with the third and fourth vibration frequencies, which

may be because the first-order frequency was more sensitive to different meshing between 2D-RPM and 3D-FEM. The 2D-RPM was more time-efficient than the 3D-FEM in analyzing the vibration modes: the 2D-RPM required 30 s with one CPU as opposed to the nearly 20 min required for the 3D-FEM with four CPUs.

**Table 1.** Comparison of the first four vibration modes and eigenvalues (Hz) of the OSFP predicted by 3D-FEM and 2D-RPM under the CCCC boundary condition.

It can be concluded that the 2D-RPM had high accuracy in free-vibration analysis of OSFP under CCCC boundary condition. To further verify the effectiveness of 2D-FEM in analyzing vibration modes, the vibration modes of OSFP under different boundary conditions are analyzed as shown in Table 2. The 3D-FEM and VAM-based 2D-RPM had good consistency in the prediction of natural frequencies and vibration modes under various boundary conditions, and the maximum error of natural frequency was less than 6% under the CCSS boundary condition. The stronger the boundary condition is, the higher the natural frequency is. The natural frequency under CCCC boundary condition was about twice that under SSSS boundary condition. The natural frequencies under CCSS and CSCS boundary conditions were almost the same, but the asymmetry of CSCS boundary condition led to the asymmetry of vibration mode.

**Table 2.** Comparison of vibration modes and natural frequencies (Hz) of the OSFP under different boundary conditions (BCs).

#### *3.4. Global Buckling Analysis*

To verify the accuracy and effectiveness of 2D-RPM, the buckling modes and critical loads of OSFP under different boundary and load conditions illustrated in Figure 11 are

listed in Table 3. The combinations of boundary and load conditions included SFFF/uniaxial (Case 1), SSFF/uniaxial (Case 2), SSSS/uniaxial (Case 3), and SSSS/biaxial (Case 4).

**Figure 11.** Four combination of boundary and load conditions used in buckling analysis.

**Table 3.** Comparison of global buckling modes and loads (N) predicted by the 2D-RPM and 3D-FEM under different boundary and load conditions.

Table 3 shows that the buckling load in Case 3 (471.83 N) was about 9 times that in Case 2 (57.96 N) and 1.577 times that in Case 4 (300.67 N). The buckling loads in Case 1 and Case 2 were basically the same. The error of the critical buckling load under various boundary conditions was less than 5%, indicating that the VAM-based 2D-RPM and 3D-FEM predictions of the global buckling of OSFP agreed closely.

#### **4. Parameter Study**

The 2D-RPM was selected to conduct parametric study. The material properties and layup configurations of the skin and the stiffeners were the same as those in Section 3, except where explicitly indicated. The boundary conditions for uniaxial buckling analysis were SFFF in Case 1, while the boundary conditions for the free-vibration analysis were CCCC.
