As shown in Figure 1A, the subsea flexible tank is balanced under buoyancy and gravity.

In the X direction, d σ cos ⁡ θ ⋅ τ ∙ y d φ − Δ P ⋅ sin ⁡ θ ⋅ y d φ d s − ρ g y d φ d s ⋅ τ = 0 , (1)

and in the Y direction, d σ sin ⁡ θ ⋅ τ ∙ y d φ + Δ P ⋅ cos ⁡ θ ⋅ y d φ d s − σ t ⋅ d φ d s ⋅ τ = 0 , (2) where Δ P = ρ s w − ρ f l d ⋅ g ⋅ x , which is the pressure difference; θ is the angle between ds and the x-axis; σ is the stress in the latitude direction; σ _t is the stress in the longitude direction; τ is the membrane thickness of the flexible tank; ρ is the equivalent density of the membrane of the flexible tank; ρ s w is the density of the seawater (1,025 kg/m³); and ρ f l d is the density of the internal fluid.

The equations can be simplified into the following: d d x σ y = ρ g y + σ t y ′ σ y = 1 y ″ − y ρ g y ′ 1 + y ′ 2 + Δ P τ 1 + y ′ 2 3 / 2 + σ t 1 + y ′ 2 , (3) When σ t = 0 , the equations become as follows: d d x σ y = ρ g y σ y = − y y ″ ρ g y ′ 1 + y ′ 2 + Δ P τ 1 + y ′ 2 3 / 2 . (4)

The boundary condition is as follows: σ x = 0 = σ 0 = G 1 + y 0 ′ 2 2 π y 0 τ y x = 0 = y 0 y ′ x = 0 = y 0 ′ , (5) where G is the load at the tank bottom, G = ρ s w − ρ f l d ⋅ g * V 0 , and V 0 is the initial volume of the tank.

The aforementioned equations can be solved using MATLAB software.

For the balloon shape, the height = 0.748 (diameter), and for pumpkin shape, the height = 0.604 (diameter) (Farley, 2005), whereas for the subsea tank, height ≈ diameter, as shown in Figure 1B. The subsea tank has a larger height/ratio than the pumpkin shape and the balloon shape.

Three prototypes were manufactured for the test, the height for two of which is 0.4 m, and for the other is 1.7 m. Each tank is made of TPU-coated aramid fabric to satisfy strength, sealing, and fluid compatibility requirements, anchored to the seabed by a mooring line and a net covering the tank surface.

The tank profile at different volumes from approximately 5%–95% volume is shown in Figure 8A for the tank with a rigid connection between the top/bottom flange, and in Figure 8B for the tank without a rigid connection. As shown in Figure 8A, the tank profile of FEA analysis during deformation is relatively similar to the tank profile observed during the test for the tank with rigid connection between the top and bottom flanges. When 95% of the fluid was inside, the tank was bulged and the tank profile was close to the natural profile. As the volume was reduced, the lower part of the tank collapsed initially during offloading, and then the collapse propagated upward. Finally, only the top part retained the initial shape when the tank was empty.

The deformed shape is due to the pressure field and the membrane stress in the fabric. According to the Young–Laplace equation, the curvature in the fabric at one point is related to the tension at that point, and the pressure is computed as follows: p = F x r x + F y r y , (6) where p is the differential pressure across the fabric; F x and F y are the tensions (per unit length) in the fabric in the directions of the principal radii of curvature; and r x and r y are the principal radii of the curvature.

When the tank is overfilled, the tank has a super-pressure status rather than a zero-pressure status, and the membrane is in a state of tension due to the internal pressure. As the volume reduces, the tank is close to a zero-pressure shape, and the latitudinal stress of the membrane is near zero. When the volume is reduced further, the bottom membrane collapses as the differential pressure leads to the inward direction. The local curvature direction changes at different points, as shown in Figure 9. The shape is a convex curve at point #1 near the outer side, whereas the shape at point #2 is a concave curve near the center line, resulting in longitudinal stress on the membrane near the bottom area. Therefore, because the tank is flexible, part of the buoyancy is transferred to the mooring line by the internal hydrostatic pressure applied on the top membrane, and part of the buoyancy is transferred to the mooring line by the rigid connection between the top flange and bottom flange. This finding can be validated by comparing the shape with/without a connection between the top and bottom flanges, as shown in Figure 8. Without a rigid connection between the top and bottom flanges, the bottom part moves upward until it touches the top side membrane; the deformed shape is similar to a jellyfish, as shown in Figure 8B.

Test results for the longitudinal strain and latitudinal strain are shown in Figure 10. The corresponding FEA strain results are shown in Figure 11. For the longitudinal membrane strain, shown in Figure 10A, the strain near the bottom (black line) increases when volume increases slightly from empty (less than 30% volume) as the buoyancy increases. Then, the strain decreases when the volume increases further. As the tank gradually expands and increases the contact surface with the mooring net from the top part, the pressure difference (buoyancy) acts on the top mooring net directly. Meanwhile, the internal pressure and outer pressure tend to balance near the bottom area. Then, the membrane is in a free state when the tank is full. The trend of the strain reverses as the volume becomes smaller from fully filled to empty. The strain trend is consistent with the deformation shown in Figure 8A. The strains in the middle and top also have a similar trend of initially increasing and then decreasing during the process with the volume changing from empty to full. However, the transition point has a larger volume, such that for the top membrane strain, the transition point is approximately 60% volume.

For the latitudinal membrane strain, shown in Figure 10B, an opposite trend can be found for the strain near the bottom area; it decreases initially and increases again when the volume increases from the empty to the full condition. The strain, when empty, is due to the local collapse deformation; then, the lower tank gradually expands when the volume increases, and the latitudinal strain decreases accordingly. A similar trend applies to the latitudinal strain at the middle and top areas. Unlike the trend of the longitudinal strain, the latitudinal strain near the top part is in a compressed state due to the constraints of the mooring net.

FEA results for the longitudinal strain are shown in Figure 11A and Table 3. The longitudinal strain is 0.71% at 95% volume, and it gradually increases to 2.38% when the volume decreases to 25%. Then, it increases to 1.38% when the volume further decreases to 5%. The longitudinal strain is less when the volume is empty or full compared with 25% volume, which is inconsistent with the trend observed during the test.

The maximum latitudinal strain located at the bottom area due to the local collapse is shown in Figure 11B. It is slightly larger than the longitudinal strain. However, it is less important because the composite membrane has no fatigue issues, and the membrane has been verified by fatigue tests, including folding-compression and tensile as one cycle load.

The maximum FEA results for the latitudinal strain and longitudinal strain are 2.76% and 2.38%, respectively, and the maximum test results for the latitudinal strain and longitudinal strain are 1.35% and 1.02%, respectively. The FEA results are conservative compared with the test results because the tank deformation is irregular, and the maximum strain is difficult to record during testing.