*2.1. Kinematics of the OSFP*

As shown in Figure 1, if the sizes of the whole panel (denoted by the macro-coordinates *xi*) are much larger than those of a unit cell (denoted by the micro-coordinates *yi*), then *yi* = *xi*/*ξ* (*ξ* is a small parameter), and the derivative of the function *f <sup>ξ</sup>* (*xα*) with respect to *xα* is

$$\frac{\partial f^{\mathbb{E}}(\mathbf{x}\_{a})}{\partial \mathbf{x}\_{a}} = \frac{\partial f(\mathbf{x}\_{a}; y\_{i})}{\partial \mathbf{x}\_{a}} \Big|\_{y\_{i} = \text{const}} + \frac{1}{\overline{\xi}} \frac{\partial f(\mathbf{x}\_{a}; y\_{i})}{\partial y\_{i}} \Big|\_{\mathbf{x}\_{a} = \text{const}} \equiv f\_{\mathcal{A}} + \frac{1}{\overline{\xi}} f\_{\mathcal{I}} \tag{1}$$

where *i*, *j* = 1, 2, 3; *α*, *β* = 1, 2.

**Figure 1.** Dimension reduction analysis of the original orthogrid-stiffened FRP panel (OSFP).

To construct a reduced-order plate model of the OSFP using VAM, the 3D displacement field of the original OSFP *ui* need to be represented by using 2D plate variables *vi* such as

$$\begin{array}{l} u\_1(\mathbf{x}\_a; y\_i) = \frac{\upsilon\_1(\mathbf{x}\_1, \mathbf{x}\_2) - \zeta y\_3 \upsilon\_{3,1}(\mathbf{x}\_1, \mathbf{x}\_2)}{\upsilon\_2(\mathbf{x}\_1, \mathbf{x}\_2) - \zeta y\_3 \upsilon\_{3,2}(\mathbf{x}\_1, \mathbf{x}\_2)} + \zeta w\_1(\mathbf{x}\_a; y\_i) \\ u\_3(\mathbf{x}\_a; y\_i) = \frac{\upsilon\_3(\mathbf{x}\_1, \mathbf{x}\_2) + \zeta w\_3(\mathbf{x}\_a; y\_i)}{\upsilon\_3(\mathbf{x}\_1, \mathbf{x}\_2) + \zeta w\_3(\mathbf{x}\_a; y\_i)} \end{array} \tag{2}$$

where *wi* is the fluctuating function to be solved, and the underlined terms should meet the following constraints

$$\begin{array}{l} w\_1 = \langle \mu\_1 \rangle + \mathfrak{F} \langle y\_3 \rangle v\_{3,1} \\ w\_2 = \langle \mu\_2 \rangle + \mathfrak{F} \langle y\_3 \rangle v\_{3,2} \\ w\_3 = \langle \mu\_3 \rangle \end{array} \tag{3}$$

where · represents the volume integration over the unit cell.

The non-underlined terms should satisfy the following conditions

$$
\langle \not g w\_i \rangle = 0 \tag{4}
$$

The 3D strain field can be expressed as

$$
\Gamma\_{ij} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial x\_j} + \frac{\partial u\_j}{\partial x\_i} \right) \tag{5}
$$

Plugging Equation (2) into Equation (5) gives the 3D strain field as

$$\begin{aligned} \Gamma\_{11} &= \varepsilon\_{11} + \overline{\zeta}y\_3\kappa\_{11} + w\_{1,1} \\ 2\Gamma\_{12} &= 2\varepsilon\_{22} + 2\overline{\zeta}y\_3\kappa\_{12} + w\_{1,2} + w\_{2,1} \\ \Gamma\_{22} &= \varepsilon\_{22} + \overline{\zeta}y\_3\kappa\_{22} + w\_{2,2} \\ 2\Gamma\_{13} &= w\_{1,3} + w\_{3,1} \\ 2\Gamma\_{23} &= w\_{2,3} + w\_{3,2} \\ \Gamma\_{33} &= w\_{3,3} \end{aligned} \tag{6}$$

where *εαβ* and *καβ* can be expressed as

$$\begin{aligned} \varepsilon\_{11}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= \upsilon\_{1,1}(\mathbf{x}\_{1},\mathbf{x}\_{2}), & \varepsilon\_{22}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= \upsilon\_{2,2}(\mathbf{x}\_{1},\mathbf{x}\_{2}),\\ 2\varepsilon\_{12}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= \upsilon\_{1,2}(\mathbf{x}\_{1},\mathbf{x}\_{2}) + \upsilon\_{2,1}(\mathbf{x}\_{1},\mathbf{x}\_{2}),\\ \kappa\_{11}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= -\upsilon\_{3,11}(\mathbf{x}\_{1},\mathbf{x}\_{2}), & \kappa\_{22}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= -\upsilon\_{3,22}(\mathbf{x}\_{1},\mathbf{x}\_{2}),\\ \kappa\_{12}(\mathbf{x}\_{1},\mathbf{x}\_{2}) &= -\upsilon\_{3,12}(\mathbf{x}\_{1},\mathbf{x}\_{2}) \end{aligned} \tag{7}$$

The 3D strain field can be obtained as

$$\begin{aligned} \Gamma\_t &= \begin{bmatrix} \Gamma\_{11} & \Gamma\_{22} & 2\Gamma\_{12} \end{bmatrix}^\mathrm{T} = \mathfrak{e} + \mathfrak{x}\_3 \mathfrak{x} + \partial\_t \mathfrak{w}\_{||} \\ \mathfrak{2}\Gamma\_5 &= \begin{bmatrix} 2\Gamma\_{13} & 2\Gamma\_{23} \end{bmatrix}^\mathrm{T} = \mathfrak{w}\_{||} + \partial\_t \mathfrak{w}\_3 \\ \Gamma\_t &= \Gamma\_{33} = \mathfrak{w}\_{3,3} \end{aligned} \tag{8}$$

where **<sup>Γ</sup>***e*, **<sup>Γ</sup>***s*, **<sup>Γ</sup>***<sup>t</sup>* are strain matrices of 3D-FEM; ()|| = [()<sup>1</sup> ()2] T, *<sup>ε</sup>* = [ *<sup>ε</sup>*<sup>11</sup> <sup>2</sup>*ε*<sup>12</sup> *<sup>ε</sup>*<sup>22</sup> ] T, *κ* = [*κ*<sup>11</sup> *κ*<sup>12</sup> + *κ*<sup>21</sup> *κ*22] T, and

$$
\partial\_t = \begin{bmatrix} \binom{\cdot}{\cdot}\_1 & 0 \\ \binom{\cdot}{\cdot}\_2 & \binom{\cdot}{\cdot}\_1 \end{bmatrix}, \partial\_t = \begin{Bmatrix} \binom{\cdot}{\cdot}\_1 \\ \binom{\cdot}{\cdot}\_2 \end{Bmatrix} \tag{9}
$$

As shown in Figure 2, the unit cell within the OSFP can be divided into three parts to facilitate the integral solution. Then we obtain the strain energy of the panel as

$$
\delta \mathcal{U} = \frac{1}{2} \int\_{-a/2}^{a/2} \int\_{-b/2}^{b/2} \frac{1}{\Omega} \mathcal{U}\_{\Omega} \mathbf{d} x\_2 \mathbf{d} x\_1 \tag{10}
$$

where

$$\begin{split} \mathcal{U}\_{\Omega} &= \int\_{-t\_1}^{0} \int\_{-\frac{t\_1}{2}}^{\frac{t\_1}{2}} \int\_{-\frac{t\_2}{2}}^{\frac{t\_2}{2}} \mathbf{F}\_A^T \mathbf{D}\_A \mathbf{T}\_A \mathrm{d}y\_1 \mathrm{d}y\_2 \mathrm{d}y\_3 \\ &+ \int\_0^h \int\_{-\frac{t\_1}{2}}^{\frac{t\_1}{2}} \int\_{-\frac{t\_2}{2}}^{\frac{t\_2}{2}} \mathbf{F}\_B^T \mathbf{D}\_B \mathbf{T}\_B \mathrm{d}y\_1 \mathrm{d}y\_2 \mathrm{d}y\_3 \\ &+ \int\_0^h \int\_{-\frac{t\_3}{2}}^{\frac{t\_3}{2}} \int\_{-\frac{t\_2}{2}}^{\frac{t\_3}{2}} \mathbf{F}\_C^T \mathbf{D}\_C \mathbf{T}\_C \mathrm{d}y\_1 \mathrm{d}y\_2 \mathrm{d}y\_3 \end{split} \tag{11}$$

with the subscripts *A*, *B*, and *C* representing the skin, longitudinal stiffener, and transverse stiffener, respectively.

**Figure 2.** Decomposition diagram of unit cell within the OSFP.

Equation (10) can be rewritten as

$$\begin{split} \mathcal{U} &= \frac{1}{2} \int\_{-a/2}^{a/2} \int\_{-b/2}^{b/2} \frac{1}{\Omega} \left< \mathbf{T}^{\mathrm{T}} \mathbf{D} \mathbf{T} \right> \mathrm{d} \mathbf{x}\_{2} \, \mathrm{d} \mathbf{x}\_{1} \\ &= \frac{1}{2} \int\_{-a/2}^{a/2} \int\_{-b/2}^{b/2} \frac{1}{\Omega} \left< \left\{ \begin{array}{cc} \mathbf{T}\_{\varepsilon} \\ 2\mathbf{T}\_{s} \\ \mathbf{\Gamma}\_{t} \end{array} \right\}^{\mathrm{T}} \begin{bmatrix} \mathbf{D}\_{\varepsilon} & \mathbf{D}\_{\mathrm{cs}} & \mathbf{D}\_{\varepsilon t} \\ \mathbf{D}\_{\varepsilon s}^{\mathrm{T}} & \mathbf{D}\_{s} & \mathbf{D}\_{st} \\ \mathbf{D}\_{t\ell}^{\mathrm{T}} & \mathbf{D}\_{st}^{\mathrm{T}} & \mathbf{D}\_{\ell} \end{bmatrix} \left\{ \begin{array}{cc} \mathbf{T}\_{\varepsilon} \\ 2\mathbf{T}\_{s} \\ \mathbf{\Gamma}\_{t} \end{array} \right\} \mathrm{d} \mathbf{x}\_{2} \, \mathrm{d} \mathbf{x}\_{1} \end{split} \tag{12}$$

where *De*, *Des*, *Det*, *Ds*, *Dst*, and *D<sup>t</sup>* are the corresponding sub-matrices of three-dimensional 6 × 6 material matrix.

The virtual work done by the applied loads is

$$\begin{array}{l} \mathbf{E} = \int\_{-b/2}^{b/2} \int\_{-a/2}^{a/2} \frac{1}{\Omega} \left( \langle f\_i u\_i \rangle + \beta\_i u\_i^- + \tau\_i u\_i^+ \right) \mathbf{dx}\_1 \mathbf{dx}\_2 \\ + \int\_{-a/2}^{a/2} \int\_{-h/2}^{h/2} \left( a\_i u\_i \right)|\_{\mathbf{x}\_2 = \pm b/2} \mathbf{dx}\_3 \mathbf{dx}\_1 + \int\_{-b/2}^{b/2} \int\_{-h/2}^{h/2} \left( a\_i u\_i \right)|\_{\mathbf{x}\_1 = \pm a/2} \mathbf{dx}\_3 \mathbf{dx}\_2 \end{array} \tag{13}$$

where *fi* is the body force, *α<sup>i</sup>* is the traction force applied on the lateral surfaces, *β<sup>i</sup>* and *τ<sup>i</sup>* denote the traction forces on the bottom and top surface, respectively.

Plugging Equation (2) into Equation (13) gives

$$\begin{array}{l} E = \int\_{-b/2}^{b/2} \int\_{-a/2}^{a/2} (p\_i v\_i + q\_a \Phi\_a) \mathrm{d}x\_1 \mathrm{d}x\_2 \\ + \int\_{-a/2}^{a/2} \int\_{-h/2}^{h/2} (P\_i v\_i + Q\_a \Phi\_a)|\_{\chi\_2 = \pm h/2} \mathrm{d}x\_3 \mathrm{d}x\_1 \\ + \int\_{-b/2}^{b/2} \int\_{-h/2}^{h/2} (P\_i v\_i + Q\_a \Phi\_a)|\_{\chi\_1 = \pm a/2} \mathrm{d}x\_3 \mathrm{d}x\_2 + E \end{array} \tag{14}$$

where Φ<sup>1</sup> = *v*3,2, Φ<sup>2</sup> = −*v*3,1, and

$$\begin{split} E^\* &= \int\_{-b/2}^{b/2} \int\_{-a/2}^{a/2} \frac{1}{\Omega} (\langle f\_l w\_i \rangle + \beta\_l w\_i^- + \tau\_l w\_i^+) \, \mathrm{d}x\_1 \mathrm{d}x\_2 \\ &+ \int\_{-a/2}^{a/2} \int\_{-h/2}^{h/2} (a\_l w\_i)|\_{\mathbf{x}\_2 = \pm b/2} \mathrm{d}x\_3 \mathrm{d}x\_1 + \int\_{-b/2}^{b/2} \int\_{-h/2}^{h/2} (a\_l w\_i)|\_{\mathbf{x}\_1 = \pm a/2} \mathrm{d}x\_3 \mathrm{d}x\_2 \end{split} \tag{15}$$

The values of *pi*, *qα*, *Pi*, and *Q<sup>α</sup>* in Equation (14) can be calculated as

$$\begin{array}{l}p\_{i} = \frac{1}{\Omega} (\langle f\_{i} \rangle + \beta\_{i} + \tau\_{i})\\q\_{1} = \frac{1}{\Omega} \left(-\mathbf{x}\_{3}^{-}\beta\_{2} - \mathbf{x}\_{3}^{+}\tau\_{2} - \langle \mathbf{x}\_{3}f\_{2} \rangle \right) \\q\_{2} = \frac{1}{\Omega} (\mathbf{x}\_{3}^{-}\beta\_{1} + \mathbf{x}\_{3}^{+}\tau\_{1} + \langle \mathbf{x}\_{3}f\_{1} \rangle) \\P\_{i} = \langle \langle \mathbf{a}\_{i} \rangle \rangle \\Q\_{1} = -\langle \langle \mathbf{x}\_{3}\mathbf{a}\_{2} \rangle \rangle \\Q\_{2} = \langle \langle \mathbf{x}\_{3}\mathbf{a}\_{1} \rangle \rangle \end{array} \tag{16}$$

According to VAM, *E*∗ can be ignored, and the total potential energy is

$$\begin{array}{l} \delta \Pi = \delta \mathcal{U} - \delta \mathcal{E} \\ = \int\_{-b/2}^{b/2} \int\_{-a/2}^{a/2} \left( \frac{1}{2} \delta \left< \mathbf{1}^{T} \bar{\mathbf{D}} \mathbf{1} \right> - p\_i \delta v\_i - q\_a \delta \Phi\_a \right) \mathbf{dx}\_1 \mathbf{dx}\_2 \\ + \int\_{-a/2}^{a/2} \int\_{-h/2}^{h/2} \left( P\_i \delta v\_i + Q\_a \delta \Phi\_a \right) \vert\_{\mathbf{x}\_2 = \pm h/2} \mathbf{dx}\_3 \mathbf{dx}\_1 \\ + \int\_{-b/2}^{b/2} \int\_{-h/2}^{h/2} \left( P\_i \delta v\_i + Q\_a \delta \Phi\_a \right) \vert\_{\mathbf{x}\_1 = \pm a/2} \mathbf{dx}\_3 \mathbf{dx}\_2 \end{array} \tag{17}$$
