## 2 Theoretical Analysis  
The main purpose of this section is to establish the non-linear control equation for the folding and bending of composite tape springs, derive the precise expressions of the z-direction displacement w and steady-state bending moment during the folding and bending of composite tape springs, analyze the factors affecting the bending moment, obtain the moment-curvature characteristic curve, and carry out finite element numerical calculations and analyses. Figure 3 is a schematic diagram of a composite tape spring, where L is the length of the cylindrical shell along the generatrix direction, R is the radius of curvature of the surface in the open shell, t is the shell thickness, and \(\theta\) is the opening angle. Figure 4 is a schematic diagram of the folding and bending of the cylindrical shell deployable mechanism, where M is the folding and bending moment, r is the curvature radius of the bending point after folding and bending, and \(\varphi\) is the opening angle of the generatrix after folding and bending. The bending and folding form is shown in Figure 4b, with a bending and folding rotation angle of and a radius of curvature of r. Establishing the coordinate system is shown in the figure, the deformation after bending and folding is shown in Figure 5a, and the analysis is carried out using the infinitesimal element. For composite structures, a cylindrical coordinate system is used. Assuming that the membrane force per unit width of the shell is N(\(N_{\varphi},N_{y},N_{\varphi y}\)), the internal bending force per unit width is M(\(M_{\varphi}\), \(M_{y}\), \(M_{\varphi y}\)), the mid-plane strain is \(\varepsilon^{0}\), and the mid-plane deflection is \(\kappa\) (\(\kappa_{\varphi},\kappa_{y\varphi}\)), the conversion is conducted. Equilibrium equation:  
\[\frac{dN_{y}}{dy}+P_{y}=0 \tag{1}\]  
\[\frac{d^{2}M_{y}}{dy^{2}}+\frac{N_{\varphi}}{r}+P_{z}+N_{y}\frac{d^{2}w}{dy^{2 }}+\frac{\partial N_{y}}{\partial y}\frac{\partial w}{\partial y}=0 \tag{2}\]  
Geometric equation:  
\[\varepsilon_{y}=\frac{dv}{dy}+z(\frac{1}{R}-\frac{d^{2}w}{dy^{2}})+\frac{1}{2 }\bigg{(}\frac{dw}{dy}\bigg{)}^{2} \tag{3}\]  
Figure 4: Schematic diagram of folding and bending of composite tape spring cylindrical shell. Figure 5: Force diagram of the infinitesimal element of folding and bending of composite tape spring. Figure 3: Schematic diagram of the geometry of composite tape spring cylindrical shell. \[\varepsilon_{\varphi}=\frac{-w}{r-z} \tag{4}\]  
Let \(K_{1}=\frac{1}{R}-\frac{d^{2}w}{dy^{2}},K_{2}=\frac{dp}{dy}+\frac{1}{2}\Big{(} \frac{dw}{dy}\Big{)}^{2}\), then  
\[\varepsilon_{y}=K_{2}+zK_{1} \tag{5}\]  
Constitutive equation:  
\[\left[ \begin{array}{c}N_{y}\\ N_{\varphi}\\ N_{\varphi y}\\ M_{\varphi}\\ M_{y}\\ M_{\varphi y}\end{array} \right]=\left[ \begin{array}{cccccccc}A_{11}&A_{12} &A_{16}&B_{11}&B_{12}&B_{16}\\ A_{12}&A_{22}&A_{26}&B_{12}&B_{22}&B_{26}\\ A_{16}&A_{26}&A_{66}&B_{16}&B_{26}&B_{66}\\ B_{11}&B_{12}&B_{16}&D_{11}&D_{12}&D_{16}\\ B_{12}&B_{22}&B_{26}&D_{12}&D_{22}&D_{26}\\ B_{16}&B_{26}&B_{66}&D_{16}&D_{26}&D_{66}\end{array} \right]\left[ \begin{array} []{c}\varepsilon_{\varphi}^{0}\\ \varepsilon_{y}^{0}\\ \gamma_{\varphi y}^{0}\\ \kappa_{\varphi}\end{array} \right] \tag{6}\]  
\[\left\{ \begin{array}{c}\sigma_{\varphi}\\ \sigma_{y}\\ \tau_{\varphi y}\end{array} \right\}=\left[ \begin{array}{cccccccc}\overline{ \mathbf{Q}_{11}}^{k}&\overline{\mathbf{Q}_{12}}^{k}&\overline{\mathbf{Q}_{16 }}^{k}\\ \overline{\mathbf{Q}_{21}}^{k}&\overline{\mathbf{Q}_{22}}^{k}&\overline{ \mathbf{Q}_{26}}^{k}\\ \overline{\mathbf{Q}_{16}}^{k}&\overline{\mathbf{Q}_{26}}^{k}&\overline{ \mathbf{Q}_{66}}^{k}\end{array} \right]\left\{\left\{ \begin{array}{c} \varepsilon_{\varphi}^{0}\\ \varepsilon_{y}^{0}\\ \gamma_{\varphi y}^{0}\\ \gamma_{\varphi y}^{0}\end{array} \right\}+\mathbf{z}\left\{ \begin{array}[] {c}\kappa_{\varphi}\\ \kappa_{y}\\ \kappa_{\varphi y}\end{array} \right\}\right\} \tag{7}\]  
Expression for internal force in the mid-surface:  
\[\left\{