*2.3. Equivalent Density of 2D-RPM*

For heterogeneous materials and structures, the equivalent density of the unit cell is usually used to characterize the weight of the reduced-order plate model. The total mass of the unit cell within the OSFP can be expressed as

$$m = m\_1 + m\_2 + m\_3 = \rho\_1 \cdot L\_1 L\_2 \cdot t\_1 + \rho\_2 \cdot L\_2 \mathbf{h} \cdot t\_2 + \rho\_3 \cdot L\_1 \mathbf{h} \cdot t\_3 \tag{32}$$

where *ρ*1, *ρ*2, and *ρ*<sup>3</sup> are the density of the skin, the longitudinal stiffener and transverse stiffener, respectively.

The total volume of the unit cell is

$$V = L\_1 L\_2 \cdot (t\_1 + h) \tag{33}$$

According to the equivalent density formula, we obtain

$$
\rho = \frac{m}{V} \tag{34}
$$

The equivalent density of the 2D-RPM can be expressed as

$$\rho^\* = \frac{\rho\_1 \cdot L\_1 L\_2 \cdot t\_1 + \rho\_2 \cdot L\_2 h \cdot t\_2 + \rho\_3 \cdot L\_1 h \cdot t\_3}{L\_1 L\_2 \cdot (t\_1 + h)} \tag{35}$$

#### **3. Validation Example**

To verify the accuracy and effectiveness of the present reduced model, the static and dynamic behaviors of OSFP predicted by the present model were compared with those of 3D-FEM. The 3D-FEM had 15 unit cells in the *x*<sup>1</sup> and *x*<sup>2</sup> direction as shown in Figure 3. The equivalent stiffness of OSFP was obtained by variational asymptotic analyzing over the unit cell shown in Figure 4b and inputted into the 2D-RPM (300 mm × 300 mm) using shell elements, as shown in Figure 4c, to analyze the static and dynamic behavior under different boundary conditions. The relative error between the 2D-RPM and 3D-FEM is defined as Error = <sup>|</sup>2D−RPM results−3D−FEM results<sup>|</sup> 3D−FEM results <sup>×</sup> 100%.

**Figure 3.** Meshing of 3D finite element model (3D-FEM).

The structural parameters shown in Figure 4a were: *l* = 20 mm, *h* = 3 mm, and *t* = 1 mm. The OSFP is made of T300/7901 carbon/epoxy laminates. The layup configuration of skin was [45/ − 45/0/ − 45/45]2s, and that of stiffener was [45/ − 45]4s. The lamina properties were: *E*<sup>11</sup> = 71.76 GPa, *E*<sup>22</sup> = *E*<sup>33</sup> = 7.81 GPa, *G*<sup>12</sup> = *G*<sup>13</sup> = 2.52 GPa, *G*<sup>23</sup> = 2.11 GPa, *v*<sup>12</sup> = *v*<sup>13</sup> = 0.343, *v*<sup>23</sup> = 0.532, *ρ* = 1.42 g/cm3. The effective plate properties of the skin and the stiffener obtained by present model were given in Figure 5 for reference.

**Figure 4.** Meshing of unit cell and 2D reduced-order plate model (2D-RPM) of OSFP.


(**a**) Skin


(**b**) Stiffener

**Figure 5.** Effective plate properties of the skin and the stiffener calculated by the present model (unit: SI).

#### *3.1. Static Displacement Analysis*

Six typical boundary conditions shown in Figure 6, including CCCC, CCSS and CSCS, CSSS, SSSS and FFCC, were used for static displacement analysis. The naming convention of boundary conditions is four letters, where S denotes simply supported constraint, C for fixed constraint, and F for free constraint.

To verify the effectiveness of the 2D-RPM, a uniform load of 5 kPa was applied to the top surface of the OSFP, and the displacement distributions along Path 1 of the 3D-FEM and 2D-RPM were compared in Figure 7. The comparative results show that the displacement distributions predicted by the 2D-RPM were basically in agreement with those of the 3D-FEM. The differences were due to the different meshing methods used in the two models. The maximum displacement error under the CCSS boundary condition was the largest, but it was still within 5%. It is worth noting that the differences between the 2D-RPM and 3D-FEM in Figure 7c–e were much greater than other cases, which may be due to the gradual enhancement of boundary constraints from SSSS to CCCC. It was concluded that the 2D-RPM can predict the static displacement of the stiffened FRP panel with high accuracy and effectiveness, and the equivalent stiffness obtained from the VAM had sufficient accuracy.

**Figure 6.** Typical boundary conditions of the OSFP.

**Figure 7.** Vertical displacement along Path 1 of the plate under a uniform load of 5 kPa and different boundary conditions.
