so that it can be confirmed that L can be regarded as point of instability.
Eq. (2.6) in its original form, considers critical load for a ring without imperfection (pcr), in fact, the collapse is reached asymptotically for the latter value. In order to give reason to the similitude in terms of buckling pressure, the comparison of a series of both numerical and theoretical simulations is needed, keeping the bending stiffness of the cross-section constant and varying the radius of the pipe. The resolution of new pcr exhibits a valid formulation which guarantees theoretical results closer to the actuals and so to the FEM outcomes, which works in terms of both displacements and collapse pressure.
The available theoretical formulation which is based on thin-wall hypothesis gives results with lower error for high values of D/t. On the contrary, the gap between theoretical and numerical outcomes rises as the ratio under analysis decreases, which is captured in Figure 2.13. Thus, from this behavior, the consequence of considering zero error for high D/t ratios and justify the following assumption for the theoretical model.
An overall of eight models are analyzed for the same carcass cross-section and material, while different D/t ratios. In Figure 2.14, the collapse loads are plotted against different geometries when they reach the ovality limit. As it was expected, the critical buckling loads decreases as the tube diameters increase. Moreover, it is possible to see the asymptotic behavior between numerical and theoretical results for growing D/t ratios.
With that being said, the hypothesis of considering zero error for D/t = 30, which is the widest geometry considered. The choice to stop the computation at that value is made because it is rare to contemplate higher diameter configurations for deep water environments. Moreover, even if the error goes to zero asymptotically, as shown in Figure 2.15, this assumption wants to be a conservative suggestion since considering it for higher D/t standards will lead to a higher collapse pressure for the case under analysis.
Once the error is eliminated, the actual collapse loads accounting for imperfections can be obtained, and their behavior is shown in Figure 2.16.
Figure 2.13 Ovality versus load for different geometries, where dashed lines stand for numerical results and continuous line stand for theoretical results.
Figure 2.14 Critical loads versus dimensionless diameters.
Figure 2.15 Error trend.
The extracted polynomial trend line depicts the guideline for the new theoretical model, and has the following formulation:
Eq. (2.17) is an estimation of the collapse pressure valid for the initial imperfection considered. It shows conservative results and it obviates the employment of the buckling load for the perfect ring, in the theoretical model, which leads to underestimation of collapse conditions.
Figure 2.16 Critical load comparison for all the model established versus dimensionless diameters.
In order to get a behavior closer to the actual, Eq. (2.7) is developed again exploiting Eq. (2.16) as theoretical derivation to get the critical buckling load, which, for the case with D = 6 in, produces pcr = 9.12 MPa. As previously done, dimensionless load and ovality are plotted for both theoretical and numerical models, as shown in Figure 2.17.
The modified theoretical model exhibits an acceptable error equal to e = 2.58%, and it obviates to the previous discrepancy simulating the behavior in terms of load in addition to displacements.
Finally, the theoretical prediction of local stresses is precise if considering pipes or rings geometry but for the stainless-steel interlocked carcass, due to the intricate profile of the cross section, while affordable results could be obtained from FEM analysis. The actual stress magnitude is not constant along the structure; in fact, dissimilar values are reached in two different points under analysis. In Figure 2.18, it is reported the relationship between stress and strain for the critical point between the two, at limit value of ovality. It is important to focus the attention on the highest stress value reached which is equal to about 400 MPa, which is far below the proportional limit stress equal to 600 MPa. Thus, it is possible to deliberate on the right assumption done for the theoretical model of considering elastic material properties only.
In this section, the results about the collapse study for the interlocked carcass are compared with the one for a steel strip reinforced thermoplastic pipes, in order to understand until which water depth there is no need of further reinforcement. The SSRTP overall design is made by a variable number of layers of thin steel strips wrapped two by two at opposite lay angles, surrounded by an inner and outer HDPE layer.
Figure 2.17 Results comparison for FEM and modified theoretical model.
Figure 2.18 Stress-strain relationship.
Figure 2.19 HDPE stress-strain relationship.
Steel strips are considered to have an elastic behavior with a modulus of Young equal to 206,000 MPa. While HDPE layers properties account for an elastic-plastic behavior, where the modulus of Young
