The composite pressure vessel is assumed to be an intermediately thick shell structure, with thickness/cylindrical radius greater than 0.18. The first-order shear deformation theory (FSDT) proposed by Mindlin [4] assumed a constant transverse shear deformation through the intermediately thick shell. The element type ?8 node Shell 281 [5] [6] used in finite element analysis is based on FSDT in this study. As shown in Figure 1, the coordinate system α-β is at the middle surface of shell, and the z-axis is in the thickness direction of shell. Assuming no normal strain, ε_z = 0, the normal of shell keeps straight during deformation, but not perpendicular to the middle surface. The displacement field can be expressed as

where u₀, v₀, w₀ represent the displacement at middle surface of shell. ψ_α and ψ_β are the mid-surface rotations.

The stress and strain relationship of k-th layer of composite laminate is shown below [7].

The coefficient of stiffness matrix () is obtained after transforming the stiffness matrix from the material principal axis to α-β axis, as shown in Equation (3) [7], in which c and s represent cosθ and sinθ respectively.

The coefficients of stiffness matrix () are expressed below [7].

where E₁₁ and E₂₂ are the Young. s modulus of composites in three directions; G₁₂, G₂₃ and G₁₃ are the shear modulus; υ₁₂, υ₂₃ and υ₁₃ are the Poisson’s ratio of composites.

In this study, the composite pressure vessel is under the operating internal pressure 20 MPa. The Tsai-Hill failure criterion was used through the thickness of the pressure vessel to determine whether the pressure vessel fails or not. The Tsai-Hill failure criterion is expressed as [8]

where σ₁ is the stress in the fiber direction, σ₂ the transverse stress and τ₁₂ the shear stress. S_L and S_T are the tensile strengths in the fiber and transverse directions and S_LT the shear strength for the composite lamina. When Equation (5) is true, it represents the possible failure at some layer in the composite pressure vessel. In the latter section, the value of left hand side of Equation (5) is termed as Tsai-Hill criterion value, which indicates failure of composite pressure vessel when this value is greater than 1.

The composite pressure vessel has overall length 808 mm with a central cylinder part 592.5 mm, as shown in Figure 2. The outside diameter of central composite section is 215.5 mm. The opening diameter of end plate is 40 mm. The thickness of composite pressure vessel is 18 mm, which is formed by a total of 144 graphite/epoxy composite layers; each layer has a thickness of 0.125 mm. The laminate stacking of composite pressure vessel is symmetrical cross-ply, [(0/90)_s]₃₆, counting from inside wall as layer 1 with fiber in the hoop direction. In Figure 3, the schematic of selected inner and outer layers is shown.

The material properties of graphite/epoxy composites are given below in Table 1 and Table 2 [9] [10]. The graphite/epoxy composite lamina is assumed to be specially orthotropic υ₁₂ = υ₁₃ =υ₂₃, E₂₂ = E₃₃, G₂₃ = E₂₂/2(1 + υ₂₃) [8], since υ₁₃, υ₂₃, E₃₃ are difficult to be measured.

The longitudinal and transverse strengths of graphite/epoxy composite lamina are adopted from [10] and the shear strength is from [9].

The operating internal pressure of composite pressure vessel is 20 MPa with compressed natural gas (CH₄) at minus 40 to 60 degrees Celsius [1] [2]. Two types of boundary conditions, clamped by buckles as the first type and free ex-

pansion as the second type, are prescribed at the junction of semispherical part in analysis, as shown in Figure 4. The two end plates are placed at both ends to restrict all axial degrees of freedom (DOFs).

The commercial finite element code ANSYS [5] was used in analysis. The composite pressure vessel is meshed using 8-node element, Shell 281 [5] [6]. The stress at a selected point at the junction of semispherical part was calculated with different element numbers. Figure 5 shows the analysis model and convergence analysis of composite pressure vessel. The difference of simulated results at selected point is less than 1% for the analysis model of composite pressure vessel with 22,400 elements and 67,456 nodes. This analysis model is used for the rest of this study.

Since the layer 1 directly contacts with the internal pressure loading, the stresses in the fiber and transverse directions for layer 1 and 2, relatively large when compared with the stresses in other layers, are of primary concern and are discussed in the following sections. Figure 7 and Figure 8 show the von Mises stress distributions of layer 1 and layer 2 of composite pressure vessel with first type and second type boundary conditions. The von Mises stress values are in the range of 80 - 100 MPa. In the cylindrical part, the stress is very high in layer 1, 163 - 180 MPa, and very small in layer 2.4.5 - 24.1 MPa in the cylindrical part

for both boundary conditions. At the junction of semispherical part, von Mises stress is 106.34 MPa for layer 1 and 98.408 MPa for layer 2 under first type boundary condition; von Mises stress is 84.64 MPa for layer 1 and 92.531 MPa for layer 2 under second type boundary condition.

The stresses in the fiber and transverse directions of composite lamina are obtained by transforming the stress components from finite element analysis in the α-β coordinate system. First the discussion for the stresses in composite pressure vessel is simply based on the maximum stress criterion. The shear stresses obtained are very small and negligible when compared with the shear strength of graphite/epoxy composites. The stresses in the fiber and transverse directions for layer 1 and layer 2 along the designated curve for first type boundary condition are shown in Figure 9. In Figure 9(a), the stress of layer 1 in the fiber direction σ₁ at the junction is 108.23 MPa, much lower than the corresponding material strength 2200 MPa; the stress in the transverse direction σ₂ is 3.89 MPa, lower than the corresponding material strength 70 MPa. Therefore no failure occurs in layer1 for the composite pressure vessel under first type boundary condition. However, in Figure 9(b), the stress of layer 2 in the fiber directionσ₁ at the junction is 4.87 MPa, much lower than the corresponding material strength 2200 MPa and the stress in the transverse direction σ₂ is 100.75 MPa, larger than the corresponding material strength 70 MPa. Therefore failure may occur in layer2 for the composite pressure vessel under first type boundary condition.

The stresses in the fiber and transverse directions for layer 1 and 2 along the designated curve for second type boundary condition are shown in Figure 10. In Figure 10(a), the stress of layer 1 in the fiber directionσ₁ at the junction is 55.915 MPa and maximum 70.692 MPa, much lower than the corresponding material strength 2200 MPa; the stress in the transverse direction σ₂ is 94.934 MPa, larger than the corresponding material strength 70 MPa. Therefore failure may occur in layer1 for the composite pressure vessel under second type boundary condition. In Figure 10(b), the stress of layer 2 in the fiber directionσ₁ at the junction is 95.99 MPa and maximum 127.51 MPa, much lower than the corresponding material strength 2200 MPa and the stress in the transverse direction σ₂ is 6.44 MPa, also lower than the corresponding material strength 70 MPa. Therefore no failure occurs in layer 2 for the composite pressure vessel under second type boundary condition.

The results from the previous section show that the stress in the transverse direction is the main reason for failure of composite pressure vessel since the stress is greater than its corresponding strength 70 MPa, based on the maximum stress criterion. The Tsai-Hill criterion, Equation (5), is also used to assess the failure of composite pressure vessel for both types of boundary conditions. Figure 11 shows the Tsai-Hill criterion value of layer 1 and 2 along the designated curve from junction. From Figure 11(a), the Tsai-Hill criterion value of layer 2 is 2.07, greater than 1; this indicates the failure of layer 2 in composite pressure vessel under first type boundary condition. On the contrary from Figure 11(b), the Tsai-Hill criterion value of layer 1 is 1.84, greater than 1; this indicates the failure of layer 1 in composite pressure vessel under second type boundary condition.