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The initial ovality is caused by manufacturing tolerances or bending loads, it must be considered at least equal to 0.002 and this can be expressed by
(2.4)
Figure 2.1 Schematic representation reference systems, initial radial displacements and initial diameters.
where
By replacing Eq. (2.2) into Eq. (2.1), it is possible to get
(2.5)
for which the solution results in
where pcr is the critical buckling pressure for a perfect ring.
The ovality is computed considering it as function of the maximum displacement uRmax. It is expressed in absolute value for cos(2) = 1; thus Eq. (2.6) assumes the following form:
Ovality shows the variation of the minor and major axes. It is computed step by step for each load increment by adding and subtracting, respectively, the magnitude of the displacements for the corresponding step from the diameters. Limit value is considered, conservatively, as 20 times the initial one and it is equal to L = 0.04. Besides this value, the pipe can be considered not suitable anymore for its purpose, being rough liner crucial elements and easily affected by turbulence due to internal flow. As it will be shown for both theoretical and numerical simulation, the pipe can be considered collapsed at L, it already exhibits large development of ovalization for almost stable pressure value.
If the pressure armor layer is not taken into account in the pipe configuration, as discussed in [1], the critical load for a perfect ring can be expressed as
where, EIeq is the equivalent ring bending stiffness of each layer per unit length of the pipe. For the interlocked carcass it is equal to
for which, n is the number of tendons in the layer, Lp is the pitch length, K is a factor that is function of the lay angle and of the moment of inertia of the section (for massive cross-section as carcass K = 1) and I2’ is the smallest moment of inertia of the cross- section which can be computed, referring to Figure 2.2, as follows:
To better control the model and to compare results, the actual load is normalized with respect to the theoretical buckling load that accounts for imperfections at its threshold L, it is equal to
Finally, the need of a reliable theoretical model suitable for practical application is used in this work, in order to evaluate the need of the interlocked carcass. As it was demonstrated in Bai et al. [3] through experimental and numerical simulations, the critical buckling load is estimated by summing up the contribution of each layer. Thus, the collapse loads for both the cross-section geometries of SSRTP considered here are computed by using the following formulation:
Figure 2.2 Carcass profile-principal outline.
for which, i and j are the number of steel and PE layers respectively. The two terms of Eq. (2.12) are derived by Eq. (2.8) for both steel and polymeric materials, as follows:
for which, n is the number of tendons in the layer, and the other parameters of Eq. (2.13) are previously mentioned. Steel strips are treated as elastic, while inner and outer PE layers are considered in plastic field. Physical non-linearities of the plastic material are expressed considering for each incremental step the update tangential modulus Ej, i, equivalent moment of inertia Ij and mean radius Rj.
In this section, the behavior of the stainless-steel carcass under external pressure is simulated using the commercial finite element software ABAQUS [4]. Finite element method (FEM)simulation is required to confirm theoretical results for the collapse behavior, thus for predicting radial displacements for each load step when the pipe is affected by hydro-static pressure. The established model is based on the pessimistic hypothesis that all the outer sheaths are damaged, and the external pressure acts directly on the interlocked carcass; thus, the latter must be designed to carry the full load. The model developed is a 3D ring model which assumes that the lay angle can be neglected. The simpler 3D ring simulation shows good agreement comparing outcomes with full 3D pipe model when the purpose is the computation of the collapse pressure for the carcass layer [5]. At the same time, this assumption reduces significantly
