Collapse of Externally Pressurized Steel–Composite Hybrid Cylinders: Analytical Solution and Experimental Verification

Abstract

To evaluate the collapse pressure of the steel–composite hybrid cylinders under external pressure without excessive computational cost, an analytical formula was derived in this study. The rationality of the derived formula was verified by the comparison with experimental and numerical results. The experimental results indicate that samples are manufactured and tested with good quality. The derived formula considered material failure and could reasonably predict the collapse pressure of the steel–composite hybrid cylinders with a maximum difference of 3.1%. Moreover, the effects of the wrap angle, thickness, and length on the collapse pressure of the hybrid cylinders were theoretically analyzed. The loading capacity of the hybrid cylinders was maximized under a wrap angle of ±55° for the composite layer. These findings are mainly because the hoop stress is twice the value of axial stress for a cylinder under uniform pressure.

1. Introduction

Steel cylindrical shells have a robust design, can be produced using mature techniques, and are widely used in ships and marine structures, such as pressure vessels and pipelines used in autonomous underwater vehicles, human-occupied vehicles, and subsea engineering applications [1,2,3,4,5]. However, steel cylinders are prone to instability or collapse under external pressure [6]. The instability of steel cylinders has been investigated analytically and experimentally [3,7,8,9,10]. Moreover, steel cylinders are sensitive to imperfections, which can considerably reduce their collapse pressure. Cylinders with imperfections thus have poor structural efficiency.

Therefore, steel–composite hybrid cylinders with improved structural efficiency have been proposed. These hybrid cylinders have attracted considerable research attention because they possess the advantages of steel and composite cylinders [11,12,13,14,15]. The inclusion of composite layers can considerably improve the structural efficiency of a steel cylinder. These layers have high specific strength and stiffness [16,17], flexible manufacturing requirements, excellent energy absorption ability [18], and high corrosion resistance [19,20]. Many studies have numerically and experimentally investigated hybrid cylinders subjected to axial and internal pressure [21,22]. Teng and Hu [21] experimentally demonstrated that external wrapping with composite layers enhanced the ductility of steel cylinders under axial compression. Subsequently, Vakili [23] used experimental and numerical approaches to demonstrate that local wrapping with composite layers substantially increases the loading capacity of an elephant-foot cylinder under axial and internal pressure. However, few studies have experimentally evaluated the collapse properties of steel–composite hybrid cylinders under external pressure.

The collapse pressure of steel and composite cylinders is typically evaluated using an analytical approach. Numerous studies have developed analytical approaches for quickly predicting the buckling or collapse pressure of steel-only cylinders. Some formulas, such as the Venstel–Krauthaer [24], Ross [25], CCS2013 (submersible specifications of the China Classification Society) [26], and ABS2012 (submersible specifications of the American Bureau of Shipping) [27] formulas, have been proposed for evaluating the loading capacity of cylinders under uniform external pressure.

For composite-only cylinders, several analytical formulas have been developed for the rapid prediction of the buckling pressure [28,29,30]. For example, Rosenow [31] developed an analytical solution based on classical lamination theory (CLT) to predict the buckling pressure of composite cylinders and reported that the filament wound pipe should be wound at 54.75° for optimal biaxial pressure loading. Imran and Shi [32,33] used linear eigenvalues to optimize the design of composite cylinders subjected to external uniform pressure. They later [34] developed an analytical model based on first-order shear deformation theory for evaluating the buckling of composite cylinders. Given multiple boundary conditions, Lopatin [35,36] predicted the critical buckling pressure of composite cylinders with various closed ends by using an analytical method. Bai [37,38,39] proposed an analytical buckling model for reinforced thermoplastic pipes that can describe cases with various load types, such as bending, tension, and external pressure. Moreover, Messager [40] proposed a linear Sanders-type buckling model of cross-ply cylinders in which winding-induced geometrical imperfections were considered.

However, no study has derived an analytical formula for predicting the collapse pressure of steel–composite hybrid cylinders under external pressure. Thus, a reasonable formula without excessive computations should be developed for predicting the collapse pressure of such cylinders.

In this study, an analytical approach and experimental approach were adopted for evaluating the collapse pressure of steel–composite hybrid cylinders under external pressure. The remainder of this paper is organized as follows: Section 2 details an analytical formula designed to evaluate the collapse pressure of steel–composite hybrid cylinders under external pressure. Section 3 describes a comparison of analytical, experimental, and numerical results for steel-only and steel–composite hybrid cylinders. This section also provides a theoretical explanation of the effects of the wrap angle, thickness, and length on the collapse pressure of such cylinders. Finally, Section 4 presents the conclusions of this study and provides suggestions for future research. The results indicate that the proposed formula can reasonably predict the collapse pressure of steel–composite hybrid cylinders. The maximum difference between the predictions and the experimental results was 3.1%. Moreover, the loading capacity of the hybrid cylinders was maximized under a wrap angle of ±55° for the composite layer.

2. Materials and Methods

An analytical formula was designed to evaluate the collapse pressure of steel–composite hybrid cylinders under external pressure. For comparison, these pressures were also calculated using the NASA SP-8700 [28] and ASME Code 2007 (RD-1172) [29] formulas on the basis of a modified stiffness matrix.

2.1. Problem Definition

A steel cylinder strengthened with composite is investigated in this study. The steel–composite hybrid cylinder comprises an outer composite layer and an inner steel layer (Figure 1). The grade of the inner steel layer is O₆Cr₁₉Ni₁₀ according to GB/T 3280-2015 [41]. This grade is equivalent to 304 according to ASTM A959-16 [42]. The carbon-fiber-reinforced polymer (CFRP) prepreg (CFS24T300) was adopted in the outer composite layer.

The two ends of the hybrid cylinder are enclosed by heavy bungs, which impose rigid boundary conditions on the cylinder’s ends [43,44]. The outer surfaces of the hybrid cylinder and the bungs are subjected to a uniform external hydrostatic pressure P₀. This loading causes the radial and axial compression of the cylinder (Figure 2).

The steel–composite hybrid cylinder has a length of L, a total thickness of h, and an inner radius of R. Moreover, the wrap angle of its composite layer is θ (Figure 3). The thicknesses of the steel and composite layers are t_s and t_c, respectively. The outer radius of the cylinder is R_s. The thickness of a single composite layer is t_l. The steel layer can be transformed into n equivalent isotropic single layers with a thickness of t_l each. Therefore, if the total number of layers is N, the numbers of steel and composite single layers are n and N − n, respectively.

2.2. Mechanical Properties

According to Hooke’s law, the stress and strain in a single layer (Figure 4) have the following relationship:

$${\epsilon }_1=\frac{{\sigma }_1}{E_1},{\epsilon }_2=-{\nu }_{12}{\epsilon }_1=-\frac{{\nu }_{12}{\sigma }_1}{E_1},{\gamma }_{12}=\frac{{\tau }_{12}}{G_{12}},$$

(1)

$${\epsilon }_2=\frac{{\sigma }_2}{E_2},{\epsilon }_1=-{\nu }_{21}{\epsilon }_2=-\frac{{\nu }_{21}{\sigma }_2}{E_2},{\gamma }_{12}=\frac{{\tau }_{12}}{G_{12}},$$

(2)

where E₁ and E₂ represent the Young’s moduli of the fiber and matrix orientations, respectively.

By substituting Equations (1) and (2) into Equation (3), the following matrix is obtained:

$$\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\gamma }_{12}\end{array}\right]=S\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\tau }_{12}\end{array}\right]=\left[\begin{array}{ccc}{S}_{11}& S_{12}& 0\\ S_{21}& S_{22}& 0\\ 0& 0& S_{66}\end{array}\right]\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\tau }_{12}\end{array}\right],$$

(3)

where $S_{11}=\frac{1}{E_1}$, $S_{22}=\frac{1}{E_2}$, $S_{12}=S_{21}=-\frac{{\nu }_{21}}{E_2}=-\frac{{\nu }_{12}}{E_1}$, and $S_{66}=\frac{1}{G_{12}}$. In isotropic materials, $S_{11}=S_{22}=\frac{1}{E}$, $S_{12}=S_{21}=-\frac{\nu }{E}$, and $S_{66}=\frac{2(1+\nu )}{E}$.

$$\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\tau }_{12}\end{array}\right]=Q\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\gamma }_{12}\end{array}\right]=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{21}& Q_{22}& 0\\ 0& 0& Q_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\gamma }_{12}\end{array}\right],$$

(4)

where $Q=S^{-1}$ represents the reduced stiffness matrix. On the basis of Equation (4), the stresses in layer k can be expressed as follows (Figure 5):

$${\left\{\sigma \right\}}_k={[\overline{Q}]}_k{\left\{\epsilon \right\}}_k,\left[\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right]=\overline{Q}\left[\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right]=\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{21}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}\end{array}\right]\left\{\left[\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right]+t_l\left[\begin{array}{c}{q}_x\\ q_y\\ q_{xy}\end{array}\right]\right\},$$

(5)

where t_l and q represent the thickness of a single layer and the curvature, respectively.

$$\begin{gathered} {\overline{Q}}_{11}={\cos }^4{\theta }_k{Q}_{11}+{\sin }^4{\theta }_k{Q}_{22}+2{\cos }^2{\theta }_k{\sin }^2{\theta }_k\left(Q_{12}+2Q_{66}\right); \\ {\overline{Q}}_{22}={\sin }^4{\theta }_k{Q}_{11}+{\cos }^4{\theta }_k{Q}_{22}+2{\cos }^2{\theta }_k{\sin }^2{\theta }_k\left(Q_{12}+2Q_{66}\right); \\ {\overline{Q}}_{66}=\left(Q_{11}+Q_{22}-2Q_{12}\right){\cos }^2{\theta }_k{\sin }^2{\theta }_k+{\left({\cos }^2{\theta }_k-{\sin }^2{\theta }_k\right)}^2{Q}_{66}; \\ {\overline{Q}}_{12}={\overline{Q}}_{21}=\left({\cos }^4{\theta }_k+{\sin }^4{\theta }_k\right)Q_{12}+\left(Q_{11}+Q_{22}-4Q_{66}\right){\cos }^2{\theta }_k{\sin }^2{\theta }_k; \\ {\overline{Q}}_{16}={\overline{Q}}_{61}=\left[{\cos }^2{\theta }_k\left(Q_{11}-Q_{12}-2Q_{66}\right)+{\sin }^2{\theta }_k\left(Q_{12}-Q_{22}+2Q_{66}\right)\right]\cos {\theta }_k\sin {\theta }_k; \\ {\overline{Q}}_{26}={\overline{Q}}_{62}=\left[{\cos }^2{\theta }_k\left(Q_{12}-Q_{22}+2Q_{66}\right)+{\sin }^2{\theta }_k\left(Q_{11}-Q_{12}-2Q_{66}\right)\right]\cos {\theta }_k\sin {\theta }_k; \end{gathered}$$

where θ_k represents the fiber orientation of the composite layer (Figure 5).

For isotropic materials, the following equations are obtained:

$$\left[\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right]=\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\tau }_{12}\end{array}\right],$$

(6)

$$\left[\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right]=\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{21}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}\end{array}\right]\left\{\left[\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right]+t_l\left[\begin{array}{c}{q}_x\\ q_y\\ q_{xy}\end{array}\right]\right\}=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& 0\\ Q_{21}& Q_{22}& 0\\ 0& 0& Q_{66}\end{array}\right]\left\{\left[\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right]+t_l\left[\begin{array}{c}{q}_x\\ q_y\\ q_{xy}\end{array}\right]\right\},$$

(7)

The equilibrium conditions applied to the force system in the composite and steel layers are presented in Figure 5 and Figure 6.

The following equilibrium equations are obtained on the basis of CLT:

$$\left\{\begin{array}{l}{N}_x\hfill \\ N_y\hfill \\ N_{xy}\hfill \end{array}\right\}={\displaystyle {\int }_{-h/2}^{h/2}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_{xy}\end{array}\right\}}dz={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{\left\{\begin{array}{c}{\sigma }_{x-s}\\ {\sigma }_{y-s}\\ {\sigma }_{zy-s}\end{array}\right\}}_k}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{\left\{\begin{array}{c}{\sigma }_{x-c}\\ {\sigma }_{y-c}\\ {\sigma }_{zy-c}\end{array}\right\}}_k}}dz,$$

(8)

$$\left\{\begin{array}{l}{M}_x\hfill \\ M_y\hfill \\ M_{xy}\hfill \end{array}\right\}={\displaystyle {\int }_{-t2}^{t/2}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_{xy}\end{array}\right\}}zdz={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{\left\{\begin{array}{c}{\sigma }_{x-s}\\ {\sigma }_{y-s}\\ {\sigma }_{zy-s}\end{array}\right\}}_{k}z}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{\left\{\begin{array}{c}{\sigma }_{x-c}\\ {\sigma }_{y-c}\\ {\sigma }_{zy-c}\end{array}\right\}}_k}}zdz,$$

(9)

where −s and −c represent the steel and composite layers, respectively; n = t_s/t_l; t_s is the thickness of the steel layer (Figure 2); h is the total thickness of the hybrid cylinder; N = n + t_c/t_l; and t_c is the thickness of the composite layer (Figure 2).

By substituting Equations (5) and (7) into Equations (8) and (9), the following equations are obtained:

$$N_x={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{11})}_k({\epsilon }_x^0+z{q}_x)+{(Q_{12})}_k({\epsilon }_y^0+z{q}_y)}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}\begin{array}{l}{({\overline{Q}}_{11})}_k({\epsilon }_x^0+z{q}_x)+\\ {({\overline{Q}}_{12})}_k({\epsilon }_y^0+z{q}_y)+\\ ({\overline{Q}}_{16})({\gamma }_{xy}^0+z{q}_{xy})\end{array}}}dz,$$

(10)

$$N_y={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{21})}_k({\epsilon }_x^0+z{q}_x)+{(Q_{22})}_k({\epsilon }_y^0+z{q}_y)}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}\begin{array}{l}{({\overline{Q}}_{21})}_k({\epsilon }_x^0+z{q}_x)+\\ {({\overline{Q}}_{22})}_k({\epsilon }_y^0+z{q}_y)+\\ ({\overline{Q}}_{26})({\gamma }_{xy}^0+z{q}_{xy})\end{array}}}dz,$$

(11)

$$N_{xy}={\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{66})}_k({\gamma }_{xy}^0+z{q}_{xy})}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}\begin{array}{l}{({\overline{Q}}_{16})}_k({\epsilon }_x^0+z{q}_x)+\\ {({\overline{Q}}_{26})}_k({\epsilon }_y^0+z{q}_y)+\\ ({\overline{Q}}_{66})({\gamma }_{xy}^0+z{q}_{xy})\end{array}}}dz,$$

(12)

By merging the resultant equation, the following equations are obtained:

$$\begin{array}{l}{N}_x=({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{11})}_k}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{11})}_k}}dz)({\epsilon }_x^0)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{12})}_k}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{12})}_k}}dz)({\epsilon }_y^0)+\\ ({\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{16})}_k}}dz)({\gamma }_{xy}^0)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{11})}_{k}z}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{11})}_k}}zdz)(q_x)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{12})}_k}}zdz+\\ {\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{12})}_k}}zdz)(q_y)+({\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{16})}_k}}zdz)(q_{xy})\\ =A_{11}({\epsilon }_x^0)+A_{12}({\epsilon }_y^0)+A_{16}({\gamma }_{xy}^0)+B_{11}(q_x)+B_{12}(q_y)+B_{16}(q_{xy})\end{array},$$

(13)

$$\begin{array}{l}{N}_y=({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{21})}_k}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{21})}_k}}dz)({\epsilon }_x^0)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{22})}_k}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{22})}_k}}dz)({\epsilon }_y^0)+\\ ({\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{26})}_k}}dz)({\gamma }_{xy}^0)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{21})}_{k}z}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{21})}_k}}zdz)(q_x)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{22})}_k}}zdz+\\ {\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{22})}_k}}zdz)(q_y)+({\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{26})}_k}}zdz)(q_{xy})\\ =A_{21}({\epsilon }_x^0)+A_{22}({\epsilon }_y^0)+A_{26}({\gamma }_{xy}^0)+B_{12}(q_x)+B_{22}(q_y)+B_{26}(q_{xy})\end{array},$$

(14)

$$\begin{array}{l}{N}_{xy}={\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{16})}_k}}dz({\epsilon }_x^0)+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{26})}_k}}dz({\epsilon }_y^0)+({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{66})}_k}}dz+\\ {\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{66})}_k}}dz)({\gamma }_{xy}^0)+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{16})}_{k}zdz}}(q_x)+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{66})}_{k}zdz}}(q_y)+\\ ({\displaystyle \sum _{k=1}^n{\displaystyle {\int }_{z_{k-1}}^{z_n}{(Q_{66})}_{k}z}}dz+{\displaystyle \sum _{k=n+1}^N{\displaystyle {\int }_{z_{k-1}}^{z_N}{({\overline{Q}}_{66})}_{k}z}}dz)(q_{xy})\end{array},$$

(15)

where A_ij and B_ij represent the extensional stiffness and bending extensional stiffness of the hybrid structure, respectively. By substituting Equations (5) and (7) into Equations (8) and (9) and merging the resultant equations, the following expression is obtained:

$$\left[\begin{array}{c}{M}_x\\ M_y\\ M_{xy}\end{array}\right]=\left[\begin{array}{llllll}{B}_{11}\hfill & B_{12}\hfill & B_{16}\hfill & D_{11}\hfill & D_{12}\hfill & D_{16}\hfill \\ B_{12}\hfill & B_{22}\hfill & B_{26}\hfill & D_{12}\hfill & D_{22}\hfill & D_{26}\hfill \\ B_{16}\hfill & B_{26}\hfill & B_{66}\hfill & D_{16}\hfill & D_{26}\hfill & D_{66}\hfill \end{array}\right]\left[\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\epsilon }_{xy}^0\\ q_x\\ q_y\\ q_{xy}\end{array}\right],$$

(16)

where D_ij is the bending stiffness of the hybrid structure and is calculated using Equations (5), (7)–(9), and (16).

2.3. Analytical Formulas

2.3.1. Designed Formula

For simultaneously considering the elastic buckling mode and material failure mode, an analytical formula was designed for calculating the collapse pressure of steel–composite hybrid cylinders under external pressure (Figure 6). The Merchant–Rankine formula can be used to determine the collapse load of a structure with a rigid support with satisfactory accuracy [45]. The collapse pressure (P_c) formula proposed in this study was derived from the measured collapse pressure of hybrid cylinder models. The Merchant–Rankine equation was marginally modified by introducing a numerical factor b. This equation comprises the elastic buckling pressure (P_e) and material failure pressure (P_f) for steel–composite hybrid cylinders. The modified Merchant–Rankine formula is expressed as follows:

$$\frac{1}{P_c}=\frac{1}{P_e}+\frac{b}{P_f},$$

(17)

where $P_c$, $P_e$, and $P_f$ represent the collapse pressure, elastic buckling pressure, and material pressure of the hybrid cylinder, respectively, and b = 0.1.

The steel–composite hybrid cylinder deforms under external pressure, and this deformation progresses until it collapses. During deformation, internal forces act on the steel and composite layers. The steel and composite materials begin to fail under the load at which the stresses match the tensile, compressive, transverse, or shear strength of the material, and the steel–composite hybrid cylinder is assumed to collapse under the failure pressure. The force resultant (Equation (18)) was determined on the basis of the constitutive equation. The stresses under external pressure can be calculated using Equations (5) and (18).

$$\{\epsilon \}={[A_{ij}]}^{-1}\{N\},$$

(18)

where N_x = −PR/2, N_y = −PR, and N_xy = 0 are the membrane forces, and P represents the applied pressure.

For each composite single layer, the total stress is separated into the fiber-directional stress (${\sigma }_{1-c}$), in-plane transversal stress (${\sigma }_{2-c}$), and in-plane shear stress (${\tau }_c$). For each steel single layer, the total stress is separated into the in-plane stress (${\sigma }_{1-s}$) and shear stress (${\tau }_s$). The ratios of these stresses to the tensile, compressive, and shear strengths can be calculated to determine the maximum failure index H. Therefore, the material failure pressure ($P_f$) of the hybrid cylinder can be obtained by dividing the applied pressure (P) by the maximum failure index (H) as follows:

$$P_f=\frac{P}{H},H=Max\left\{\frac{\left|{\sigma }_{1-c}\right|}{X_T},\frac{\left|{\sigma }_{1-c}\right|}{X_C},\frac{\left|{\sigma }_{2-c}\right|}{Y_T},\frac{\left|{\sigma }_{2-c}\right|}{Y_C},\frac{\left|{\tau }_c\right|}{S_c},\frac{\left|{\sigma }_{1-c}\right|}{T_s},\frac{\left|{\sigma }_{1-c}\right|}{C_s},\frac{\left|{\sigma }_{1-c}\right|}{S_s}\right\},$$

(19)

where $X_T$, $X_C$, $Y_T$, $Y_C$, and $S_c$ represent the fiber tensile, fiber compressive, matrix tensile, matrix compressive, and shear strengths of the composite layer, respectively. Moreover, $T_s$, $C_s$, and $S_s$ represent the tensile, compressive, and shear strengths of the steel layer, respectively.

The proposed analytical model for elastic buckling ($P_e$) does not consider transverse shear effects. The presented linear buckling analysis is based on the principles of the Messager [40] model. Equilibrium is achieved between the internal and external forces acting in the cylindrical coordinate system on a shell element. The following three partial differential equations are satisfied:

$$\frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=0,$$

(20)

$$\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_y}{\partial y}=0,$$

(21)

$$\frac{{\partial }^2{M}_x}{\partial x^2}+2\frac{{\partial }^2{M}_{xy}}{\partial x\partial y}+\frac{{\partial }^2{M}_y}{\partial y^2}-\frac{N_y}{R}+N_x^0\frac{{\partial }^2\omega }{\partial x^2}+N_y^0\frac{{\partial }^2\omega }{\partial y^2}=0,$$

(22)

where $N_x^0$ and $N_y^0$ represent the membrane forces.

According to thin shell theory, the strain–displacement relationship [40,46] is expressed as follows:

$$\left[\begin{array}{c}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial }{\partial x}& 0& 0\\ 0& \frac{\partial }{\partial y}& \frac{1}{R}\\ \frac{\partial }{\partial y}& \frac{\partial }{\partial x}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right],\left[\begin{array}{c}{q}_x\\ q_x\\ q_{xy}\end{array}\right]=\left[\begin{array}{ccc}0& 0& -\frac{{\partial }^2}{\partial x^2}\\ 0& \frac{\partial }{R\partial y}& -\frac{{\partial }^2}{\partial y^2}\\ 0& \frac{\partial }{R\partial x}& -2\frac{{\partial }^2}{\partial x\partial y}\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right],$$

(23)

Moreover, the displacement approximation functions indicating the shape of the instability mode of the cylindrical shell can be expressed as presented in Equation (24). The cylindrical shell subjected to uniform external pressure is simply supported at edges x = 0 and x = L.

$$\{\begin{array}{l}u=U\cos \alpha x\cos \beta y\hfill \\ v=V\sin \alpha x\sin \beta y\hfill \\ \omega =W\sin \alpha x\cos \beta y\hfill \end{array},$$

(24)

where α = mπ/L and β = n/R; m and n are the numbers of longitudinal and circumferential half waves, respectively; R is the inner radius of the cylindrical shell; and U, V, and W are unknown constants that represent the maximum displacements in the x-direction, y-direction, and z-direction, respectively.

By substituting Equations (13)–(16), (23), and (24) into Equations (20)–(22) and integrating the governing equation, an eigenvalue problem with a simple form (Equation (25)) is obtained. The lowest P_e value is the elastic buckling pressure of the hybrid structure.

$$P_e=\frac{P_{em}}{F_C},([K]+P_{em}[L])\left[\begin{array}{c}U\\ V\\ W\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],$$

(25)

where $K_{11}=A_{11}{\alpha }^2+A_{66}{\beta }^2$, $K_{12}=K_{21}=\left(A_{12}+A_{66}\right)\alpha \beta$, $K_{13}=K_{31}=\frac{A_{12}}{R}\alpha +B_{11}{\alpha }^3+\left(B_{12}+2B_{66}\right)\alpha {\beta }^2$, $K_{22}=A_{22}{\beta }^2+A_{66}{\alpha }^2$, $K_{23}=K_{32}=\left(B_{12}+2B_{66}\right){\alpha }^2\beta +\frac{A_{22}}{R}\beta +B_{22}{\beta }^3$, $K_{33}=D_{11}{\alpha }^4+2\left(D_{12}+2D_{66}\right){\alpha }^2{\beta }^2+D_{22}{\beta }^4+\frac{A_{22}}{R^2}+\frac{2B_{12}}{R}{\alpha }^2+\frac{2B_{22}}{R}{\beta }^2$, $L_{11}=L_{12}=L_{13}=L_{21}=L_{22}=L_{23}=L_{31}=L_{32}=0$, and $L_{33}=-\frac{R}{2}{\alpha }^2-R{\beta }^2$. The parameters A_ij, B_ij, and D_ij are computed using Equations (13)–(16), and F_C is the safety factor.

2.3.2. Modified NASA SP-8700 Formula

The NASA SP-8700 formula [28], which is based on CLT, can be used to determine the buckling pressure of a composite cylinder. By using the modified stiffness matrix (Section 2.2), this formula (Equation (26)) can be used to calculate the collapse pressure of a steel–composite hybrid cylinder.

$$P_{NASA}^{mo}=Min(P_{amn}),P_{amn}=\frac{R}{F_{NASA}\left[n^2+\frac{1}{2}{\left(\frac{m\pi R}{L}\right)}^2\right]}\frac{\det \left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]}{\det \left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]},$$

(26)

where $C_{11}=A_{11}{\left(\frac{m\pi }{L}\right)}^2+A_{66}{\left(\frac{n}{R}\right)}^2$, $C_{22}=A_{22}{\left(\frac{n}{R}\right)}^2+A_{66}{\left(\frac{m\pi }{L}\right)}^2$, $C_{33}=D_{11}{\left(\frac{m\pi }{L}\right)}^4+2\left(D_{12}+2D_{66}\right){\left(\frac{m\pi }{L}\right)}^2{\left(\frac{n}{R}\right)}^2+D_{22}{\left(\frac{n}{R}\right)}^4+\frac{A_{22}}{R^2}+\frac{2B_{22}}{R}{\left(\frac{n}{R}\right)}^2+\frac{2B_{12}}{R}{\left(\frac{m\pi }{L}\right)}^2$, $C_{13}=C_{31}=\frac{A_{12}}{R}\left(\frac{m\pi }{L}\right)+B_{11}{\left(\frac{m\pi }{L}\right)}^3+\left(B_{12}+2B_{66}\right)\left(\frac{m\pi }{L}\right){\left(\frac{n}{R}\right)}^2$, $C_{12}=C_{21}=\left(A_{12}+A_{66}\right)\left(\frac{m\pi }{L}\right)\left(\frac{n}{R}\right)$, $C_{23}=C_{32}=\left(B_{12}+2B_{66}\right){\left(\frac{m\pi }{L}\right)}^2\left(\frac{n}{R}\right)+\frac{A_{22}}{R}\left(\frac{n}{R}\right)+B_{22}{\left(\frac{n}{R}\right)}^3$.

The parameters R and L represent the inner radius and length of the cylinder, respectively, and m and n are integers necessary to produce the correct minimum pressure. The term F_NASA represents the safety factor (1~5). The C matrices are computed using Equations (13)–(16).

2.3.3. Modified ASME Code 2007 (RD-1172) Formula

The ASME Code 2007 formula [29] (Equation (27)) describes the buckling pressure of a composite cylinder and is used to estimate the allowable pressure. By using the modified stiffness matrix of the hybrid cylinder, Equation (27) can be used to calculate the collapse pressure as follows:

$$P_{ASME}^{mo}=\frac{(KD)0.8531\gamma E_{hf}^{3/4}{E}_{at}^{1/4}{t}^{5/2}}{{\left(1-v_x{v}_y\right)}^{3/4}L{\left(R\right)}^{3/2}{F}_{ASME}},$$

(27)

where $v_x=\frac{-{\left(AB{D}^{-1}\right)}_{5,4}}{{(ABD)}_{4,4}}$, $v_y=\frac{-{\left(AB{D}^{-1}\right)}_{5,4}}{{\left(AB{D}^{-1}\right)}_{5,5}}$, $E_{af}=\frac{12}{t^3{\left(AB{D}^{-1}\right)}_{4,4}}$, $E_{hf}=\frac{12}{t^3{\left(AB{D}^{-1}\right)}_{5,5}}$, and $E_{at}=\frac{A_{11}{A}_{22}-A_{12}^2}{A_{22}t}$. The matrices A, B, and D are computed using Equations (13)–(16), KD is the knockdown factor (0.84), and γ is the reduction factor. If Z_p ≤ 100, $\gamma =1-0.001Z_p$. If Z_p > 100, γ = 0.9. Furthermore, t is the wall thickness, and F_ASME is the safety factor. The parameter Z_p is calculated as follows:

$$Z_p=\frac{E_{hf}{}^{3/2}{E}_{at}{}^{1/2}}{E_{af}^2}{\left(1-v_x{v}_y\right)}^{1/2}\frac{L^2}{\left(Rt\right)},$$

(28)

3. Results and Discussion

This section presents a comparison of the analytical and experimental results obtained for steel-only and steel–composite hybrid cylinders. The elastic buckling pressures of the hybrid cylinders were calculated using analytical formulas and compared with numerical results. Moreover, the effects of the wrap angle, thickness, and length on the collapse pressure of hybrid cylinders were theoretically analyzed.

3.1. Verification of the Analytical Model

3.1.1. Steel-Only Cylinders

Steel–composite hybrid cylinders comprise an outer composite layer and an inner steel layer. The steel layer can be transformed into n equivalent isotropic single layers (Section 2). To verify the rationality of this transformation, the calculated collapse pressure of steel-only cylinders (t_c = 0) was compared with the experimental results.

Table 1. Collapse pressure (MPa) of steel-only cylinders (hybrid cylinder with t_c = 0) obtained using analytical formulas and in experiment I.

Case I [47]	${\mathit{P}}_{\mathit{C}\mathit{C}\mathit{S}}$	${\mathit{P}}_{\mathit{A}\mathit{B}\mathit{S}}$	${\mathit{P}}_{\mathit{t}\mathit{e}\mathit{s}\mathit{t}}$ [47]	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
							CCS-Test	ABS-Test	C-Test
L/R = 1.0	4.500	3.600	4.800	5.029	4.228	4.898	0.063	0.250	0.020
L/R = 1.5	3.520	2.450	3.270	3.446	2.670	3.386	0.076	0.251	0.035
L/R = 2.0	2.900	1.870	2.970	2.805	2.003	2.765	0.024	0.370	0.069

L/R = ratio of the cylinder length to the inner radius; P_CCS and P_ABS were calculated using the CCS2013 [26] and ABS2012 [27] formulas, respectively; CCS-test = |P_CCS − P_test|/P_test; ABS-test = |P_ABS − P_test|/P_test; C-test = |P_C − P_test|/P_test.

presents the collapse pressure of steel-only cylinders obtained using the proposed formula (P_C), using the CSS2013 formula (P_CCS) [26], using the ABS2012 formula (P_ABS) [27], and in the experiments (P_test) of Zhu [47]. The safety factor (F_C) was 2.8. The nominal radius and thickness of the steel-only cylinders were 50 and 1 mm, respectively, in the experiments. In Table 1, L/R indicates the ratio of the cylinder length to the cylinder radius.

Table 2. Collapse pressure (MPa) of the steel-only cylinders (hybrid cylinder with t_c = 0) obtained using analytical formulas and in experiment II.

Case II [6]	${\mathit{P}}_{\mathit{V}\text{-}\mathit{K}}$	${\mathit{P}}_{\mathit{R}\mathit{o}\mathit{s}\mathit{s}}$	${\mathit{P}}_{\mathit{t}\mathit{e}\mathit{s}\mathit{t}}$ [6]	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
							V-K-Test	Ross-Test	C-Test
SL1	4.802	4.949	4.621	4.833	3.850	4.747	0.039	0.071	0.027
SL2	4.958	5.113	4.683	4.916	3.912	4.827	0.059	0.092	0.031

P_V-K and P_Ross were calculated using the Venstel–Krauthaer [24] and Ross [25] formulas, respectively; V-K-test = |P_V-K − P_test|/P_test; Ross-test = |P_Ross − P_test|/P_test; C-test = |P_C − P_test|/P_test.

presents the collapse pressure of steel-only cylinders obtained using the derived formula (P_C) (Equation (17)), using the Venstel–Krauthaer formula ($P_{V\text{-}K}$ (Equation (29)) [24], using the Ross formula ($P_{Ross}$) (Equation (30)) [25], and in the experiments (P_test) of Zhang [6]. The safety factor was 2.8. The nominal radius, length, and thickness of the steel-only cylinders (SL) were 35, 130, and 1 mm, respectively, in the experiments.

$$P_{V\text{-}K}=0.92\frac{Et}{l}{\left(\frac{t}{r}\right)}^{1.5},$$

(29)

$$P_{Ross}=\frac{2.6E{\left(\frac{t}{2r}\right)}^{2.5}}{\frac{l}{2r}-0.45{\left(\frac{t}{2r}\right)}^{0.5}},$$

(30)

The constitutive material used in the calculations was the same as that adopted in the corresponding experiments [6,47]. Table 1 and Table 2 reveal that the proposed formula determined the collapse pressure of the steel-only cylinders with acceptable accuracy in both cases. The errors were 0.020–0.069 and 0.027–0.031 for verification examples I and II, respectively. The minimum errors were 0.020 and 0.027, which are lower than those of the other analytical formulas.

3.1.2. Steel–Composite Hybrid Cylinders

To verify the rationality of the analytical model for the collapse pressure of a steel–composite hybrid cylinder, the [55/−55]₄ and [90/90/0/90/90/0/90/90] orientations were chosen as examples of test validation. Five steel–composite hybrid cylinders comprising outer carbon-fiber-reinforced polymer (CFRP) layers and an inner steel layer were fabricated. The flowchart of sample fabrication is depicted in Figure 7a. The hybrid cylinders were enclosed by O-rings acting as heavy steel bungs. The thickness of the steel bungs was 30 mm, and the nominal thickness (t_s) and outer radius (R_s) of the steel layer and cylinders were 1.5 and 79.5 mm, respectively. The Young’s modulus and Poisson’s ratio of the steel were 190 GPa and 0.3, respectively. These heavy steel bungs imposed relatively rigid boundary conditions on the cylinder ends. The hand lay-up method was adopted to fabricate the steel–composite hybrid structure. To achieve excellent bonding, the outer surface of the steel cylinder was abrasively blasted and then cleaned using acetone. The steel cylinder was then wrapped using eight CFRP prepreg (CFS24T300) layers (Jiangsu Boshi Carbon Fiber Technology). The nominal thickness (t_l) of the one-ply prepreg was 0.15 mm. The mechanical properties of CFRP were provided by the manufacturer and were as follows: E₁₁ = 115 GPa, E₂₂= 7.72 GPa, G₁₂ = 3.72 GPa, Nu₁₂ = 0.33, X_T = 1400.09 MPa, X_C = 580.06 MPa, Y_T = 44.36 MPa, Y_C = 133.03 MPa, and S_C = 45.04 MPa. Three hybrid cylinders exhibited wrap sequences of [55/−55]₄, where 90° represents the circumferential direction of the cylinder, and were denoted as CYH1, CYH2, and CYH3. Two hybrid cylinders had wrap sequences of [90/90/0/90/90/0/90/90] and were denoted as CYL1 and CYL2. The lengths of the CYH and CYL cylinders were 320 and 280 mm, respectively. Each specimen was varnished using polyurea coating GDJN001 to prevent water absorption [48]. The polyurea-coated films had a total thickness of 0.25 mm. The polyurea layers were applied at 5 h intervals under a temperature of <30 °C.

The testing chamber was an internal cylindrical pressure vessel (Figure 7c) located at Jiangsu University of Science and Technology, Zhenjiang, China. The inner diameter, total inner height, and maximum pressure of the testing chamber were 200 mm, 1000 mm, and 8 MPa, respectively. Each sample was submerged in water without any constraints. A manual water pump was used to apply pressure. The destruction of each sample was accompanied by a loud noise and a decrease in pressure. After the flanges were removed using a heat gun (Figure 7d), the collapse modes of the five samples were observed (Figure 8). The collapse modes appear as local dents. Such local dents are associated with the initial geometric imperfections and plasticity of the steel layer. These results indicate that the collapse mode of such a hybrid structure is similar to the typical characteristic of the shell of revolution under uniform external pressure. As the composite layer has poor ductility, fiber breakage generally occurred due to dent deformation of the steel layer. The fiber breakage of CYH occurs near its center, as depicted in Figure 8a. These failure modes were mainly caused by large deformation in the center. Because of such deformation, the crack extended to the edge of the local dent first in the wrapped direction and then in the opposite direction.

The history of pressure applied to the cylinders, obtained from the hydrostatic tests, is depicted in Figure 9. Each applied pressure monotonically increased until it reached its maximum value, which is the collapse pressure of the corresponding sample. Each pressure fluctuation resembles a staggered slope because of the manual discontinuous operation of the water pump. The staggered horizontal and upward small segments indicate pressure holding and application processes, respectively. Once the maximum value is reached, the pressure on each cylinder suddenly decreases, suggesting the collapse of the cylinders. This decrease occurs because the volume of the steel cylinder is reduced due to deformation. That reveals the plasticity deformation of the steel layer occurred.

The experiments were highly repeatable. The collapse pressures of CYH1, CYH2, and CYH3 were 2.920, 2.819, and 2.820 MPa, respectively. The average collapse pressure and the maximum difference between the collapse pressure were 2.853 MPa and 3.46%, respectively. The collapse pressures of CYL1 and CYL2 were 3.232 and 2.964 MPa, with the average of and difference between these values being 3.098 MPa and 8.29%, respectively. On the basis of the modified stiffness matrix, the NASA SP-8700, ASME Code 2007 (RD-1172), and proposed formulas were used to calculate the collapse pressure of the hybrid cylinders with F_C = 2.8 (

Table 3. Experimental and analytical collapse pressure (MPa) of the steel–composite hybrid cylinders (t_c-nominal/t_s-nominal = 0.8).

Sample	θ (°)	${\mathit{P}}_{\mathit{t}\mathit{e}\mathit{s}\mathit{t}}$	${\mathit{P}}_{\mathit{t}\mathit{e}\mathit{s}\mathit{t}-\mathit{a}\mathit{v}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
							NASA-Test	ASME-Test	C-Test
CYH1	[±55]4	2.920	2.853	3.581	2.448	2.805	0.255	0.142	0.017
CYH2		2.819
CYH3		2.820
CYL1	[90/90/ 0/90/90 /0/90/90]	3.232	3.098	4.241	2.798	3.194	0.369	0.097	0.031
CYL2		2.964

θ = wrap angle of the outer composite layer of a hybrid cylinder.

). The NASA SP-8700 and ASME Code 2007 formulas exhibited errors of 25.5–36.9% and 9.7–14.2%, respectively. Although the error of the ASME Code 2007 formula was smaller than that of the NASA SP-8700 formula, both errors were unacceptably large. The proposed formula accounts for the interaction between the elastic buckling mode and the material failure mode. The average differences between the predictions of the proposed formula and the experimental results were 1.7% and 3.1% for CYH and CYL, respectively. Therefore, the proposed formula predicts collapse pressure for hybrid cylinders more accurately than the NASA SP-8700 and ASME Code 2007 formulas.

Nonlinear RIKS analysis was performed using the ABAQUS/Standard software to reveal the mechanism of collapse behavior of the samples. The finite element model of the sample was established in accordance with the sample’s real geometric shape, which was determined from scanning data. The multilayer shell-type approach was adopted to avoid interface problems associated with hybrid cylinders. This modeling approach has been successfully used to examine the collapse properties of steel–composite cylinders under axial or internal pressure [12,13,15]. As depicted in Figure 10, first, shell elements (S4R) present on the outer surface of the steel layer are assigned an outward offset section. The continuum shell elements (SC8R) are then generated using a mesh offset in the ABAQUS environment. The thickness of the inward offset is the nominal thickness of the CFRP layers (t_c = (N − n) × t_l). The elements of the steel and CFRP layers share the same nodes and are connected with each other (Figure 10). Two perfect surfaces are drawn and connected to the two ends of each cylinder to simulate the two heavy bungs. The elements of the cylinder and bungs share the same nodes; that is, they are connected. A unit of external pressure is applied to the external surface of each finite-element model. To prevent rigid body displacement, three-point-constraint boundary conditions were used in the analysis. These conditions do not overconstrain the problem because the pressure is equally applied. CCS2018 [49] recommends using similar boundary conditions for evaluating the buckling of various revolution shells under uniform external pressure, and these conditions have been widely used to evaluate the buckling of pressure shells [5].

For the case of the cylinder CYL1, the numerical results correlate satisfactorily with the experimental data. The numerically obtained equilibrium curve is presented in Figure 11. The curve shows an initial linear increase before a gradual decrease. The maximum von Mises equivalent stress of the steel layer at the maximum pressure point was equal to the material yield point, which suggests that nonlinear elastic–plastic buckling of the steel layer occurred under uniform external pressure. The collapse pressure of the cylinder CYL1, as determined using nonlinear RIKS analysis, is 3.024 MPa, 6.64% lower than that in the experimental data. The difference between the predictions of the derived formula and the numerical result was 5.32% for CYL1. Therefore, the derived formula could reasonably predict the collapse pressure of the hybrid cylinder. Moreover, the postcollapse mode at the end of the pressure curves in Figure 11 formed a local dent. These findings are identical to the experimental observations displayed in Figure 8b. Such a local dent is associated with the initial geometric imperfections and plasticity of the steel layer.

For further comparison, a linear eigenvalue analysis was performed using the ABAQUS/Standard software to examine the buckling pressure of the steel–composite hybrid cylinders, and the obtained numerical results were compared with the analytical results. The elastic buckling pressure of steel–composite hybrid cylinders were analytically calculated using the modified NASA SP-8700, ASME Code 2007, and proposed formulas for F_C = 1. The length (L) and radius (R_s) of the cylinders were 320 and 80 mm, respectively. The ratio of the composite layer to the steel layer was 0.2, 0.8, or 1.6. The wrap sequences of the hybrid cylinders were [15/−15]₄, [30/−30]₄, [45/−45]₄, [50/−50]₄, [55/−55]₄, [60/−60]₄, [65/−65]₄, [75/−75]₄, and [85/−85]₄. Elastic buckling modes obtained using ABAQUS are listed in

Table 4. Elastic buckling modes obtained using ABAQUS.

tc/ts	θ (°)
	15	30	45	50	55	60	65	75	85
0.2	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4
0.8	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4	n = 4
1.6	n = 4	n = 4	n = 3	n = 3	n = 3	n = 3	n = 3	n = 3	n = 3

n = circumferential waves.

. The analytical and numerical results for the buckling pressure of these hybrid cylinders are listed in

Table 5. Numerical and analytical buckling pressure (MPa).

tc/ts	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	${\mathit{P}}_{\mathit{a}\mathit{b}\mathit{a}\mathit{q}\mathit{u}\mathit{s}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}/{\mathit{P}}_{\mathit{a}\mathit{b}\mathit{a}\mathit{q}\mathit{u}\mathit{s}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}/{\mathit{P}}_{\mathit{a}\mathit{b}\mathit{a}\mathit{q}\mathit{u}\mathit{s}}$	${\mathit{P}}_{\mathit{C}}/{\mathit{P}}_{\mathit{a}\mathit{b}\mathit{a}\mathit{q}\mathit{u}\mathit{s}}$
0.2	15	4.266	1.640	2.627	2.865	1.489	0.382	0.917
	30	4.407	1.658	2.823	2.918	1.511	0.379	0.968
	45	4.641	1.745	3.015	3.039	1.527	0.383	0.992
	50	4.730	1.791	3.047	3.091	1.531	0.386	0.986
	55	4.820	1.842	3.059	3.145	1.533	0.391	0.973
	60	4.905	1.896	3.056	3.200	1.533	0.395	0.955
	65	4.983	1.948	3.042	3.252	1.532	0.399	0.935
	75	5.104	2.036	3.007	3.343	1.527	0.406	0.900
	85	5.167	2.085	2.985	3.397	1.521	0.409	0.879
0.8	15	6.106	2.175	3.805	3.798	1.608	0.382	1.002
	30	7.333	2.346	4.465	4.337	1.691	0.361	1.030
	45	9.344	3.018	5.422	5.397	1.731	0.373	1.005
	50	9.672	3.324	5.546	5.813	1.664	0.381	0.954
	55	9.957	3.638	5.611	6.223	1.600	0.390	0.902
	60	10.192	3.938	5.636	6.608	1.542	0.397	0.853
	65	10.374	4.207	5.641	6.951	1.492	0.403	0.812
	75	10.600	4.617	5.640	7.471	1.419	0.412	0.755
	85	10.689	4.832	5.640	7.744	1.380	0.416	0.728
1.6	15	11.639	3.966	6.441	6.308	1.845	0.419	1.021
	30	15.872	4.543	7.962	8.440	1.881	0.359	0.943
	45	20.627	6.455	9.796	11.866	1.738	0.363	0.826
	50	22.039	7.331	10.207	12.405	1.777	0.394	0.823
	55	23.225	8.185	10.429	12.882	1.803	0.424	0.810
	60	24.161	8.962	10.535	13.629	1.773	0.438	0.773
	65	24.850	9.630	10.597	13.631	1.823	0.471	0.777
	75	25.603	10.601	10.687	14.123	1.813	0.500	0.757
	85	25.693	11.089	10.715	14.375	1.787	0.514	0.745

, which also presents the wrap angles for these cylinders. The ratios of the buckling pressure calculated with the NASA SP-8007, ASME Code 2007, and proposed formulas to the numerical pressure values were 1.380–1.881, 0.359–0.514, and 0.745–1.030, respectively. For the proposed formula, the average ratio between the calculated and numerical pressure was 0.9, with the standard deviation being 0.09 and the standard error being 0.029 (Equation (31) [50]). Therefore, the proposed formula can accurately evaluate the buckling pressure of hybrid cylinders and is superior to the other two formulas in this task.

$$S{E}^2=\frac{{\displaystyle \sum {\left(\frac{P_C}{P_{abaqus}}-1\right)}^2}}{N(N-1)},$$

(31)

3.2. Effects of the Wrap Angle, Thickness, and Length on the Collapse Pressure

The modified NASA SP-8700 ($P_{NASA}^{mo}$), modified ASME Code 2007 ($P_{ASME}^{mo}$), and proposed formula (P_C) were used to determine the effects of the length, wrap angle, and number of layers on the collapse pressure of hybrid cylinders with F_C = 2.8. The length (L)-to-radius (R_s) ratios of the cylinders were 2, 3, 4, 6, 8, and 10. The ratios between the thicknesses of the composite (t_c) and steel (t_s) layers were 0.2, 0.4, 0.8, 1.2, and 1.6. The wrap angle (θ) of the cross-ply composite layer of the hybrid cylinders was 15°–85°. The analytical results for the collapse pressure of these cylinders are listed in

Table 6. Collapse pressure (MPa) of hybrid cylinders obtained using analytical formulas when t_c/t_s = 0.2.

$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference		$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
					NASA-C	ASME-C					NASA-C	ASME-C
2	15	3.073	2.187	2.120	0.450	0.032	6	1.029	0.729	0.894	0.151	0.185
	30	3.189	2.211	2.268	0.406	0.025		1.060	0.737	0.934	0.135	0.211
	45	3.312	2.327	2.392	0.385	0.027		1.110	0.776	0.984	0.129	0.211
	50	3.348	2.388	2.407	0.391	0.008		1.130	0.796	0.998	0.132	0.202
	55	3.381	2.457	2.409	0.404	0.020		1.149	0.819	1.011	0.137	0.190
	60	3.412	2.528	2.401	0.421	0.053		1.168	0.842	1.021	0.144	0.175
	65	3.439	2.597	2.388	0.440	0.088		1.185	0.866	1.029	0.152	0.159
	75	3.477	2.714	2.357	0.475	0.151		1.212	0.905	1.040	0.166	0.130
	85	3.496	2.781	2.339	0.495	0.189		1.226	0.927	1.045	0.174	0.113
3	15	2.061	1.458	1.584	0.302	0.079	8	0.807	0.547	0.722	0.118	0.242
	30	2.139	1.474	1.681	0.272	0.123		0.837	0.553	0.757	0.107	0.269
	45	2.262	1.552	1.792	0.263	0.134		0.894	0.582	0.810	0.104	0.281
	50	2.287	1.592	1.805	0.267	0.118		0.917	0.597	0.828	0.107	0.279
	55	2.312	1.638	1.812	0.276	0.096		0.940	0.614	0.845	0.112	0.274
	60	2.334	1.685	1.812	0.288	0.070		0.964	0.632	0.861	0.119	0.266
	65	2.354	1.731	1.809	0.301	0.043		0.986	0.632	0.875	0.126	0.278
	75	2.385	1.809	1.799	0.326	0.006		1.016	0.679	0.892	0.139	0.240
	85	2.400	1.853	1.791	0.340	0.034		1.020	0.695	0.891	0.144	0.220
4	15	1.524	1.094	1.246	0.223	0.122	10	0.645	0.437	0.590	0.094	0.258
	30	1.574	1.106	1.311	0.200	0.157		0.662	0.442	0.610	0.084	0.275
	45	1.657	1.164	1.390	0.192	0.163		0.687	0.465	0.637	0.080	0.269
	50	1.689	1.194	1.411	0.197	0.154		0.697	0.478	0.645	0.081	0.259
	55	1.721	1.228	1.428	0.206	0.140		0.707	0.491	0.652	0.084	0.246
	60	1.752	1.264	1.441	0.216	0.123		0.716	0.505	0.658	0.088	0.232
	65	1.780	1.299	1.449	0.228	0.104		0.725	0.519	0.663	0.093	0.217
	75	1.823	1.357	1.459	0.249	0.070		0.738	0.543	0.670	0.101	0.190
	85	1.845	1.390	1.463	0.261	0.050		0.745	0.556	0.674	0.105	0.175

,

Table 7. Collapse pressure (MPa) of hybrid cylinders obtained using analytical formulas when t_c/t_s = 0.4.

$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference		$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
					NASA-C	ASME-C					NASA-C	ASME-C
2	15	3.456	2.453	2.575	0.342	0.047	6	1.133	0.784	1.018	0.112	0.230
	30	3.742	2.456	2.818	0.328	0.129		1.216	0.809	1.099	0.106	0.264
	45	4.039	2.731	3.077	0.313	0.112		1.355	0.910	1.226	0.105	0.258
	50	4.136	2.882	3.138	0.318	0.081		1.407	0.961	1.234	0.140	0.222
	55	4.223	3.045	3.179	0.328	0.042		1.458	1.015	1.270	0.148	0.201
	60	4.299	3.209	3.206	0.341	0.001		1.507	1.070	1.303	0.157	0.179
	65	4.359	3.363	3.223	0.353	0.044		1.550	1.121	1.329	0.166	0.157
	75	4.434	3.613	3.241	0.368	0.115		1.615	1.204	1.367	0.181	0.119
	85	4.464	3.751	3.249	0.374	0.154		1.647	1.250	1.385	0.189	0.097
3	15	2.285	1.567	1.863	0.226	0.159	8	0.880	0.588	0.792	0.111	0.258
	30	2.492	1.618	2.046	0.218	0.209		0.960	0.607	0.868	0.106	0.301
	45	2.720	1.821	2.247	0.211	0.190		1.110	0.683	1.000	0.111	0.317
	50	2.790	1.922	2.297	0.215	0.163		1.170	0.721	1.048	0.117	0.312
	55	2.855	2.030	2.336	0.222	0.131		1.188	0.761	1.060	0.120	0.282
	60	2.912	2.139	2.366	0.231	0.096		1.203	0.802	1.069	0.125	0.250
	65	2.961	2.242	2.389	0.240	0.061		1.217	0.841	1.076	0.131	0.219
	75	3.029	2.409	2.421	0.251	0.005		1.236	0.903	1.086	0.139	0.168
	85	3.061	2.500	2.436	0.256	0.026		1.246	0.938	1.090	0.143	0.140
4	15	1.683	1.176	1.442	0.167	0.185	10	0.710	0.470	0.651	0.090	0.278
	30	1.819	1.213	1.569	0.159	0.227		0.755	0.485	0.697	0.083	0.303
	45	2.046	1.366	1.766	0.158	0.227		0.827	0.546	0.764	0.082	0.285
	50	2.131	1.441	1.831	0.164	0.213		0.854	0.576	0.787	0.085	0.267
	55	2.215	1.523	1.889	0.172	0.194		0.880	0.609	0.808	0.089	0.246
	60	2.292	1.604	1.940	0.182	0.173		0.905	0.642	0.827	0.094	0.224
	65	2.361	1.682	1.982	0.191	0.152		0.927	0.673	0.843	0.099	0.202
	75	2.449	1.807	2.035	0.203	0.112		0.960	0.723	0.867	0.108	0.166
	85	2.462	1.875	2.041	0.206	0.081		0.977	0.750	0.879	0.112	0.146

,

Table 8. Collapse pressure (MPa) of hybrid cylinders obtained using analytical formulas when t_c/t_s = 0.8.

$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference		$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
					NASA-C	ASME-C					NASA-C	ASME-C
2	15	4.697	3.113	3.206	0.465	0.029	6	1.445	0.906	1.190	0.215	0.239
	30	5.484	3.335	3.704	0.480	0.100		1.715	1.043	1.399	0.225	0.255
	45	6.506	4.024	4.326	0.504	0.070		2.171	1.341	1.734	0.252	0.226
	50	6.820	4.432	4.473	0.525	0.009		2.338	1.477	1.842	0.270	0.198
	55	7.086	4.850	4.568	0.551	0.062		2.499	1.617	1.935	0.292	0.164
	60	7.202	5.250	4.584	0.571	0.145		2.645	1.750	2.012	0.315	0.130
	65	7.226	5.609	4.560	0.584	0.230		2.770	1.870	2.073	0.336	0.098
	75	7.212	6.156	4.512	0.599	0.364		2.813	2.052	2.083	0.350	0.015
	85	7.182	6.443	4.484	0.602	0.437		2.812	2.148	2.077	0.353	0.034
3	15	3.011	2.016	2.080	0.447	0.031	8	1.122	0.716	0.962	0.167	0.255
	30	3.552	2.106	2.422	0.467	0.130		1.371	0.782	1.162	0.180	0.327
	45	4.245	2.683	2.843	0.493	0.057		1.669	1.006	1.398	0.194	0.280
	50	4.474	2.955	2.951	0.516	0.001		1.738	1.108	1.447	0.200	0.234
	55	4.679	3.234	3.026	0.546	0.068		1.800	1.213	1.488	0.210	0.185
	60	4.850	3.500	3.075	0.577	0.138		1.856	1.313	1.520	0.221	0.137
	65	4.984	3.739	3.106	0.605	0.204		1.903	1.402	1.546	0.231	0.093
	75	5.146	4.104	3.137	0.641	0.308		1.967	1.539	1.580	0.245	0.026
	85	5.204	4.295	3.146	0.654	0.365		1.997	1.611	1.596	0.251	0.009
4	15	2.181	1.450	1.647	0.324	0.119	10	0.894	0.573	0.789	0.133	0.274
	30	2.619	1.564	1.949	0.344	0.197		1.042	0.626	0.917	0.137	0.317
	45	3.337	2.012	2.405	0.388	0.163		1.285	0.805	1.118	0.149	0.280
	50	3.454	2.216	2.470	0.399	0.103		1.373	0.886	1.185	0.158	0.252
	55	3.556	2.425	2.513	0.415	0.035		1.458	0.970	1.246	0.170	0.221
	60	3.640	2.625	2.540	0.433	0.034		1.535	1.050	1.298	0.183	0.191
	65	3.705	2.805	2.556	0.450	0.097		1.602	1.122	1.341	0.194	0.163
	75	3.786	3.078	2.573	0.471	0.196		1.699	1.231	1.402	0.212	0.122
	85	3.818	3.221	2.580	0.480	0.249		1.746	1.289	1.432	0.219	0.100

,

Table 9. Collapse pressure (MPa) of hybrid cylinders obtained using analytical formulas when t_c/t_s = 1.2.

$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference		$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
					NASA-C	ASME-C					NASA-C	ASME-C
2	15	6.627	4.062	4.302	0.540	0.056	6	1.936	1.232	1.672	0.158	0.263
	30	8.332	4.611	5.183	0.607	0.110		2.519	1.416	2.128	0.184	0.335
	45	10.317	5.983	6.239	0.654	0.041		3.518	1.994	2.877	0.223	0.307
	50	10.634	6.734	6.387	0.665	0.054		3.825	2.245	3.087	0.239	0.273
	55	10.839	7.481	6.442	0.683	0.161		3.894	2.494	3.127	0.245	0.203
	60	10.938	8.173	6.437	0.699	0.270		3.943	2.724	3.149	0.252	0.135
	65	10.953	8.777	6.404	0.710	0.371		3.978	2.926	3.162	0.258	0.075
	75	10.838	9.277	6.326	0.713	0.466		4.015	3.222	3.176	0.264	0.015
	85	10.710	9.667	6.276	0.707	0.540		4.027	3.374	3.182	0.266	0.060
3	15	4.154	2.669	3.103	0.339	0.140	8	1.525	0.924	1.356	0.124	0.318
	30	5.207	2.983	3.774	0.380	0.210		2.003	1.062	1.748	0.146	0.392
	45	6.766	3.989	4.736	0.429	0.158		2.493	1.496	2.153	0.158	0.305
	50	7.266	4.490	4.997	0.454	0.101		2.652	1.684	2.275	0.166	0.260
	55	7.547	4.987	5.116	0.475	0.025		2.797	1.870	2.378	0.176	0.214
	60	7.604	5.448	5.116	0.486	0.065		2.922	2.043	2.462	0.187	0.170
	65	7.621	5.851	5.100	0.494	0.147		3.025	2.194	2.529	0.196	0.132
	75	7.594	6.445	5.064	0.500	0.273		3.166	2.417	2.620	0.208	0.078
	85	7.557	6.747	5.043	0.499	0.338		3.229	2.530	2.662	0.213	0.050
4	15	2.982	1.961	2.398	0.243	0.182	10	1.173	0.739	1.071	0.096	0.310
	30	3.927	2.141	3.053	0.286	0.299		1.495	0.850	1.348	0.109	0.370
	45	4.980	2.992	3.786	0.315	0.210		2.032	1.197	1.800	0.129	0.335
	50	5.251	3.367	3.953	0.328	0.148		2.224	1.347	1.953	0.139	0.310
	55	5.481	3.740	4.075	0.345	0.082		2.406	1.496	2.089	0.151	0.284
	60	5.665	4.086	4.159	0.362	0.017		2.569	1.635	2.206	0.165	0.259
	65	5.804	4.388	4.216	0.376	0.041		2.709	1.755	2.304	0.176	0.238
	75	5.962	4.834	4.282	0.392	0.129		2.909	1.933	2.441	0.191	0.208
	85	6.014	5.061	4.306	0.397	0.175		3.001	2.024	2.505	0.198	0.192

and

Table 10. Collapse pressure (MPa) of hybrid cylinders obtained using analytical formulas when t_c/t_s = 1.6.

$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	θ (°)	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference		$\frac{\mathit{L}}{{\mathit{R}}_{\mathit{s}}}$	${\mathit{P}}_{\mathit{N}\mathit{A}\mathit{S}\mathit{A}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{A}\mathit{S}\mathit{M}\mathit{E}}^{\mathit{m}\mathit{o}}$	${\mathit{P}}_{\mathit{C}}$	Difference
					NASA-C	ASME-C					NASA-C	ASME-C
2	15	9.368	5.407	5.679	0.650	0.048	6	2.648	1.696	2.237	0.184	0.242
	30	12.515	6.368	7.017	0.783	0.092		3.693	1.942	2.999	0.231	0.353
	45	15.332	8.702	8.416	0.822	0.034		5.131	2.869	4.024	0.275	0.287
	50	15.920	9.775	8.665	0.837	0.128		5.324	3.258	4.159	0.280	0.217
	55	16.282	10.913	8.753	0.860	0.247		5.476	3.638	4.247	0.289	0.144
	60	16.432	11.949	8.742	0.880	0.367		5.590	3.983	4.303	0.299	0.074
	65	16.414	12.841	8.691	0.889	0.478		5.674	4.280	4.340	0.307	0.014
	75	16.102	14.134	8.573	0.878	0.649		5.771	4.712	4.389	0.315	0.073
	85	15.779	14.786	8.487	0.859	0.742		5.807	4.929	4.412	0.316	0.117
3	15	5.684	3.572	4.077	0.394	0.124	8	2.129	1.225	1.855	0.148	0.339
	30	7.612	4.159	5.156	0.477	0.193		2.767	1.456	2.358	0.173	0.383
	45	10.182	5.738	6.587	0.546	0.129		3.681	2.152	3.075	0.197	0.300
	50	10.524	6.516	6.775	0.553	0.038		3.978	2.444	3.290	0.209	0.257
	55	10.746	7.275	6.855	0.568	0.061		4.245	2.728	3.468	0.224	0.213
	60	10.865	7.966	6.869	0.582	0.160		4.476	2.987	3.611	0.240	0.173
	65	10.905	8.560	6.857	0.590	0.248		4.666	3.210	3.725	0.253	0.138
	75	10.856	9.423	6.818	0.592	0.382		4.923	3.534	3.881	0.269	0.090
	85	10.779	9.857	6.793	0.587	0.451		5.036	3.696	3.952	0.274	0.065
4	15	4.157	2.644	3.227	0.288	0.181	10	1.570	0.980	1.416	0.109	0.307
	30	5.669	3.029	4.184	0.355	0.276		2.147	1.165	1.893	0.134	0.384
	45	7.367	4.303	5.281	0.395	0.185		3.123	1.721	2.675	0.167	0.357
	50	7.871	4.887	5.567	0.414	0.122		3.468	1.955	2.933	0.182	0.334
	55	8.295	5.457	5.768	0.438	0.054		3.792	2.183	3.159	0.200	0.309
	60	8.629	5.975	5.902	0.462	0.012		4.082	2.390	3.350	0.219	0.287
	65	8.875	6.420	5.995	0.480	0.071		4.212	2.568	3.430	0.228	0.251
	75	9.144	7.067	6.101	0.499	0.158		4.102	2.827	3.352	0.224	0.157
	85	9.176	7.393	6.119	0.500	0.208		4.045	2.957	3.315	0.220	0.108

. The error of the ASME formula, which was calculated using the expression $(P_{ASME}^{mo}-P_{test})/P_{test}$, was lower than that of the NASA formula for L/R_S values of 2, 3, and 4. The collapse pressures calculated with the ASME and proposed formulas were similar for medium-length and short cylinders.

The calculation results were obtained using the proposed formula for the effects of the wrap angle (θ) on the collapse pressure of the steel–composite hybrid cylinders with t_c/t_s values of 0.2, 0.8, and 1.6 (Figure 12). When t_c/t_s = 0.2, the collapse pressure of the hybrid cylinders was similar for θ values of 15°–85° (Figure 12a). For a t_c/t_s value of 0.8 or 1.6, as θ increased, the collapse pressure initially increased and then stabilized (Figure 12a,b). Therefore, the increase in θ can improve the collapse pressure, which can be increased by 113.09% (t_c/t_s = 1.6, L/R_S = 2). Notably, the inflection points of the trends occur at θ = 55°–65°. The collapse pressure was maximized when L/R_S = 2 and θ = 55°. The aforementioned results suggest that a wrap angle of ±55° can optimize the loading capacity of steel–composite hybrid cylinders. This angle is the classical wrap angle for a composite pressure shell and is widely used for composite cylinders subjected to external pressure [51,52]. These findings are mainly because the hoop stress is twice the axial stress of a cylinder subjected to external pressure.

For each length, the collapse pressure monotonically increased with an increase in t_c/t_s (t_s = 1.5 mm, Figure 13). Notably, the rate of this increase reduced as L/R_S increased and was affected by the wrap angle (θ). That is because the loading capacity of the longer cylinders considerably decreased. The thickness increasing plays a weak part in improving the loading capacity of the longer cylinders. For an L/R_S value of 2, the maximum increases in the collapse pressure with the t_c/t_s value were 167.86% and 262.93% when θ was 15° and 85°, respectively. The collapse pressure also decreased monotonically as L/R_S increased (Figure 14). This decrease was initially rapid but then became more gradual. These findings are mainly because the collapse resistance is considerably higher for shorter cylinders. The length plays a prominent part in improving the loading capacity of the shorter cylinders. The aforementioned trends are reasonably consistent with those obtained for steel-only cylinders [47].

4. Conclusions

To evaluate the collapse pressure of the steel–composite hybrid cylinders without excessive computational cost, a derived analytical formula was presented in the present work. The rationality of the derived formula was verified by the comparison with experimental and numerical results. The CYH and CYL hybrid cylinder comprises an outer composite layer and an inner steel layer. The wrap sequences of the composite layer were [55/−55]₄ and [90/90/0/90/90/0/90/90] for CLH and CLY, respectively. Moreover, the effects of the wrap angle, thickness, and length on the collapse pressure of the hybrid cylinders were theoretically analyzed. The following primary conclusions were drawn:

(1)

The derived formula can determine the collapse pressure of steel-only cylinders with acceptable accuracy. The errors of this formula with respect to the results obtained in two verification experiments were 0.020–0.069 and 0.027–0.031. The minimum errors of the proposed formula in these experiments were 0.020 and 0.027, which were lower than the corresponding errors of other analytical formulas.

(2)

The experimental results obtained for the steel–composite hybrid cylinders were repeatable. The maximum difference between the experimental collapse pressure was 8.29%. These findings indicate that samples are manufactured and tested with good quality. The average difference between the collapse pressure calculated using the proposed formula and the experimental results was 1.7% and 3.1% for the CYH and CYL cylinders, respectively. The derived formula considered material failure and could reasonably predict the collapse pressure of the steel–composite hybrid cylinders.

(3)

An increase in wrap angle caused an increase in the collapse pressure (by up to 113.09%). Notably, the inflection points of the trends occur at a wrap angle ranging from 55° to 65°. The maximum collapse pressure was observed. The aforementioned results suggest that the loading capacity of steel–composite hybrid cylinders can be maximized under a wrap angle of ±55°. These findings are mainly because the hoop stress is twice the value of axial stress for cylinders under uniform pressure.

(4)

For each cylinder length, the collapse pressure monotonically increased as the thickness ratios of the composite layer to the steel layer increased. However, the rate of this increase was reduced as the length-to-radius ratio increased and was considerably affected by the wrap angle. That is because the loading capacity of the longer cylinders considerably decreases. The thickness increasing plays a weak part in improving the loading capacity of the longer cylinders. Moreover, as the length-to-radius ratio increased, the collapse pressure initially decreased rapidly and then decreased gradually. These findings are mainly because the collapse resistance is considerably higher for shorter cylinders. The length plays a prominent part in improving the loading capacity of the shorter cylinders.

Future studies should examine the effects of local damage propagation on the collapse properties of steel–composite hybrid cylinders by conducting hydrostatic tests and developing corresponding analytical models for these cylinders.