Structural capacity analysis of a vertical connection module using an isotropic hardening model under external loading

ABSTRACT

The Vertical Connection Module (VCM) is widely used in the subsea connection of flexible pipelines across the Brazilian coast, and it is among one of the most important pieces of equipment applied for enabling the petroleum mixture, i.e., water, rock particulates, and hydrocarbons, to flow by Vertical Christmas Tree (VXT), Pipeline End Terminal (PLET), Pipeline End Manifold (PLEM), and Manifold up to the offshore platform facilities, such as Floating Production Storage and Offloading (FPSO) vessel. In this work, a sensitivity analysis was performed by Finite Element Method (FEM) considering the application of external loading through a flexible pipeline on the upper flange of a illustrative VCM; tension, shear, and bending moment were individually applied to verify their influence on the structural capacity of the equipment in question using the isotropic hardening model from ASME BPVC.VIII.2 (Annex 3-D) to develop the stress-strain monotonic curves of the materials. The positive and negative directions of the loads were also investigated across the plastic strain levels inferred in step 1, within the interval [0; 2%]. Focusing on the loading magnitude, the VCM exhibited worst mechanical behavior when the pure shear load was applied. Regarding the isotropic hardening model, it was observed that by increasing the plastic strain level in step 1, the VCM capacity was reduced in the subsequent loading cycle, i.e., step 2. An exception to this phenomenology was only observed for small plastic strain levels under negative tension and negative shear loads. The isotropic hardening model enabled a finer understanding of the influence of tension, shear, and bending moment on the VCM structural capacity. These observations can enhance design robustness and establish more reliable constraints for the VCM, ultimately contributing to a safer operational scenario for the Oil & Gas industry in the subsea connection systems.

Keywords:
Plasticity; Isotropic hardening model; Finite element analysis; Vertical connection module

1. INTRODUCTION

1.1. Oil & gas industry perspective

The VCM is one of the most versatile subsea equipment applied in Brazil’s Oil & Gas industry to ensure the connection between the Subsea Production System (SPS) and flexible pipelines [¹]. This equipment presents a high risk of catastrophic failure in the field due to severe loading conditions from offshore vessels – imposed by environmental parameters, e.g., wind, oceanic current, and irregular waves – in addition to the loads transferred from flexible pipeline connection to its swivel flange during subsea installation [², ³]. Due to this undesirable failure, material and financial losses, irrecoverable environmental impacts, and extremely negative media exposure may occur for the companies involved and their associated legal liabilities [⁴]. Although the VCM design and manufacturing costs are lower than those of the well-known VXT and Manifold, its structural failure can negatively affect these subsea assets, requiring them to be recovered and replaced with another equipment to restore the integrity of the SPS. This operation, conducted by specialized vessels, may require several weeks or even many months to be fully completed; in addition to costing millions of dollars [⁵, ⁶].

Nowadays, there is still no consensus in the Oil & Gas industry regarding the analysis methodology for more accurately evaluating the structural capacity of VCMs. Furthermore, investigating the structural response of the VCM under external loading conditions – tension, shear, and bending moment – is an essential step for defining its design and operational constraints. This step also serves as a sensitivity analysis, providing insights into how each external load influences the equipment’s strength. An alternative to mitigate operational risks throughout the VCM lifecycle, and, additionally, to optimize its design, is to establish a robust analysis methodology that incorporates best engineering practices and reliable industry standards.

Another analysis with significant engineering relevance is to calculate the structural capacity of the VCM for small and successive loading and unloading steps, considering a level of plasticity inferred from previous stage to determine whether its strength has been derated or not. In this case, the isotropic hardening model can be extremely useful for capturing its mechanical behavior.

In this work, a VCM will be analyzed through an FEM model considering the application of external loads, using an isotropic hardening model for its materials as defined by ASME BPVC.VIII.2 (Annex 3-D) [⁷]. A successive loading-unloading-loading steps will be performed, named step 1 and step 2, for distinct levels of plastic strain inferred from the first loading phase, i.e., step 1, with the aim of determining whether the structural capacity of the VCM is improved or reduced; in other words, whether its mechanical strength is enhanced or derated, respectively. The acceptance criterion applied is based on the elastic-plastic analysis by [⁸]. It is important to note that evaluating the VCM in only two successive loading steps is a simplified representation of the real operation scenario in the field. However, these stages are representative and sufficient to demonstrate the purpose of the academic research presented herein.

1.2. Vertical connection module

The VCM includes at least the following components in its design: a guide system, a diverless connector, a flange, a gooseneck pipe, a lifting structure, and a swivel. All these components are applicable only to annulus and production VCMs. In the case of an umbilical VCM, the gooseneck pipe and swivel are not used, as their sole purpose is to provide an electric and hydraulic supplies through the power umbilical. Therefore, any VCM design includes a similar set of components as described above, although in many cases they are manufactured with different material specifications and geometric configurations.

The VCM installation sequence can be classified as either the 1^st DVC or the 2^nd DVC, or simply as first ends or second ends. The term DVC refers to Direct Vertical Connection. This classification allows us to easily determine which flexible pipe termination will be installed first on subsea permanent equipment, such as the VXT, Manifold, and others. For the 1^st DVC operation, the VCM lands on the vessel table so that the flexible pipeline can be connected to its swivel-flange using threaded studs and heavy nuts. The lower face of the diverless hydraulic connector is free to translate and rotate in all Degrees of Freedom (DoFs), meaning that the locking segments remain unlocked. Thereafter, the flexible pipeline is gradually unreeled from the PLSV, and the bend stiffener, vertebra, buoys, and dead weight, if required, are connected to the pipe system. When the VCM approaches to the permanent equipment coordinates (azimuth), the PLSV winch is actuated to align the VCM, through its guide system, with the hub-mandrel of the permanent equipment. This action is defined as verticalization step of the installation process. After that, the control fluid in the soft-landing device chamber of the aforementioned equipment is drained, and the diverless hydraulic connector is actuated by an ROV to lock the VCM onto the hub-mandrel profile and ensure a watertight connection by energizing, i.e., local plastically deformed, a metal seal.

Figure 1 presents a VCM with funnel-up guide system [⁹]. Note the yellow-dark funnel (Base Adaptadora de Produção, BAP) positioned upward to the left, and the gray cylinder inside it, next to the operator. The picture illustrates VXT, BAP, and the annulus and production VCMs used in the Pre-Salt project.

Figure 1
VCM with funnel-up guide system [⁹].

After completing the 1^st DVC, the remaining flexible pipeline ends will be connected to the new VCM. This VCM is now connected in a configuration referred to as the second ends. Thus, for the 2^nd DVC operation, the VCM is landed on the vessel table with the second ends of the flexible pipeline, while the first ends are brought up from the previous VCM already connected on subsea. After the aforementioned step, the VCM is relocated to another subsea permanent equipment to ensure the inter-well connection or to establish the tie-in between a well and the stationary production unit, e.g., platform, FPSO, etc. Considering that the VCMs are correctly installed on both ends, a new phase of commissioning can begin with hydrostatic testing, followed by subsea production or injection, depending on the objective of the drilled well. The hydrostatic test is an essential phase that ensures the system’s tightness before starting the well production or injection.

1.3. Isotropic hardening model

For an elementary constitutive model of stress-strain behavior with isotropic hardening, the strength surface for ductile materials expands uniformly in the radial direction from the initial yield surface. This surface expands after each cycle of loading and unloading, passing through the material’s yield strength point. Under monotonic loading conditions, which is the case addressed in this work, the isotropic hardening model exhibits the same behavior as the kinematic hardening model [¹⁰, ¹¹]. If the material is subjected to subsequent loading-unloading cycles, the resulting strength surface may expand beyond the initial yield surface. This increase in material strength capacity – arising internally from intrusion and extrusion phenomena – is referred to in the literature as cold hardening [¹²]. The inverse of this effect – when the subsequent yield surface is reduced relative to the initial yield surface – is defined as softening [¹³, ¹⁴], which is more common at elevated temperatures and/or high loading rates [¹⁵].

The elastic stress-strain relationship, as stated in Hooke’s law, is presented in Equation (1) [¹⁰].

(1) $$\sigma =E\left(\epsilon -{\epsilon }^p\right)$$

To formulate strain-hardening plasticity, the expansion, i.e., the hardening effect, will be represented by Eσ and must satisfy the following conditions [¹⁰]:

1. The hardening is isotropic and the Eσ remains at the origin;

2. The hardening depends on the amount of plastic flow, i.e., $\left|{\epsilon }^{{}^{\cdot }p}\right|$;

3. The condition i leads to the yield criterion expressed in Equation (2).

(2) $$f\left(\sigma ,\alpha \right)=\left|\sigma \right|-\left({\sigma }_Y+\Gamma \alpha \right)\le 0\therefore \text{α}\ge \text{0}$$

where Γ is the plastic modulus, and σ_Y > 0 and Γ ≥ 0 are given constants. If and Γ < 0, softening occurs, as previously discussed. The α: [0, Ω] → ℝ can be defined as the internal hardening variable, and it is non-negative number.

Condition ii takes a simple evolutionary form, as expressed in Equation (3).

(3) $$\dot{\alpha }=\left|{\epsilon }^{{}^{\cdot }p}\right|$$

Equation (4) presents the consequence of the irreversible mechanism governing plastic flow, as established by the flow rule in Equation (3).

(4) $${\epsilon }^{{}^{\cdot }p}=\gamma \sin \left(\sigma \right)\therefore \gamma \ge 0$$

The irreversible mechanism based on the plastic flow rule was postulated by Kuhn-Tucker, considering both loading and unloading conditions:

(5) $$\gamma \ge 0,f\left(\sigma ,\alpha \right)\le 0,\gamma f\left(\sigma ,\alpha \right)=0$$

where γ≥ 0 is obtained from the consistency condition expressed in Equation (6).

(6) $$\gamma \dot{f}\left(\sigma ,\alpha \right)=0$$

Moreover, stress-strain curves can be generated through using various constitutive models, each governed by its flow rule. Regarding pressure vessels, the ASME BPVC.VIII.2 (Annex 3-D) covers a stress-strain model for isotropic hardening of ductile materials like carbon steels. This semiempirical model is formulated through mathematical equations, well established in the technical literature, and it is presented by Equations (7–19) [⁷].

(7) $${\epsilon }_t=\frac{{\sigma }_t}{E_y}+{\gamma }_1+{\gamma }_2$$

(8) $${\gamma }_1=\frac{{\epsilon }_1}{2}[1.0-\tanh (H)]$$

(9) $${\gamma }_2=\frac{{\epsilon }_2}{2}[1.0+\tanh (H)]$$

(10) $${\epsilon }_1={\left(\frac{{\sigma }_t}{A_1}\right)}^{\frac{1}{m_1}}$$

(11) $$A_1=\frac{{\sigma }_{ys}(1+{\epsilon }_{ys})}{{[\ln (1+{\epsilon }_{ys})]}^{m_1}}$$

(12) $$m_1=\frac{\ln (R)+({\epsilon }_p-{\epsilon }_{ys})}{\ln \left[\frac{\ln (1+{\epsilon }_p)}{\ln (1+{\epsilon }_{ys})}\right]}$$

(13) $${\epsilon }_2={\left(\frac{{\sigma }_t}{A_2}\right)}^{\frac{1}{m_2}}$$

(14) $$A_2=\frac{{\sigma }_{uts}\exp (m_2)}{m_2{}^{m_2}}$$

(15) $$H=\frac{2\{{\sigma }_t-[{\sigma }_{ys}+K({\sigma }_{uts}-{\sigma }_{ys})]\}}{K({\sigma }_{uts}-{\sigma }_{ys})}$$

(16) $$R=\frac{{\sigma }_{ys}}{{\sigma }_{uts}}$$

(17) $${\epsilon }_{ys}=0.002$$

(18) $$K=1.5R^{1.5}-0.5R^{2.5}-R^{3.5}$$

As described in [⁶], the stress-strain curve shall be limited to true UTS at true ultimate strain. After this point, the isotropic hardening model must be established as perfectly plastic, and the true Ultimate Tensile Strength (UTS) must be at true ultimate tensile, and strain must be calculated as presented in Equation (19).

(19) $${\sigma }_{uts,t}={\sigma }_{uts}\exp \left(m_2\right)$$

The parameters m₂ and ε_p shall be calculated from Table 1 [⁷]; the manufacturing materials of the VCM analyzed in this study were classified as Ferritic Steel.

Table 1
Stress-strain curve parameters [⁷].

MATERIAL (CATEGORY)	TEMPERATURE LIMIT	m2	ɛp
Ferritic Steel	480ºC (900ºF)	0.60 (1.00-R)	2.0E-5
Stainless Steel and Nickel Base Alloys	480ºC (900ºF)	0.75 (1.00-R)	2.0E-5
Duplex Stainless Steel	480ºC (900ºF)	0.70 (0.95-R)	2.0E-5
Precipitation-Hardening, Nickel-Based Austenitic Alloys	540ºC (1,000ºF)	1.09 (0.93-R)	2.0E-5
Aluminum	120ºC (250ºF)	0.52 (0.98-R)	5.0E-6
Copper	65ºC (150ºF)	0.50 (1.00-R)	5.0E-6
Titanium and Zirconium	260ºC (500ºF)	0.50 (0.98-R)	2.0E-5

All the parameters defined by Equations (7–19) will be presented in the Section 2.1.2 for both manufacturing materials considered in the VCM design, such as the gooseneck pipe and flanges.

1.4. State of the art

The structural analysis of VCMs has not been widely addressed in the scientific literature since the beginning of their utilization in the 1980s, and this research effort is far less expressive than what is observed for pipelines and riser systems in the Oil & Gas industry. This gap is even more evident in Brazilian offshore applications, particularly in deep and ultra-deep waters fields where the Pre-Salt reservoirs are located. The stronger emphasis on the riser systems is more closely related to product cost than to the risks and consequences associated with SPS premature failure, into which the VCMs are fully integrated.

As discussed previously, the VCM can be defined as a pressure vessel, and its construction has specific subsystems responsible for many functions – focusing here in the annulus and production VCMs – e.g., withstanding the external loads from flexible pipelines, providing effective sealing between the flowline and swivel flange using a metal seal, ensuring the hydraulic connector remains functional, and, consequently, the VCM locked on subsea permanent equipment during its whole lifecycle in the field, providing opening or closing of valves through ROV panel, etc. Furthermore, there are few scientific publications regarding the VCMs in the literature, and only a small part of them is truly dedicated to analyzing the mechanical behavior of this equipment when subjected to an external loads – e.g., tension, shear, and bending moment – from flexible pipelines.

Concerning the main industry standards under which the VCMs can be designed and structurally analyzed, we can cite API STD 17G [¹⁶], API STD 6X [¹⁷], ASME BPVC.VIII.2 [⁷], ASME BPVC.VIII.3 [¹⁸], and ISO 13628-7 [⁸]. These industry standards, although some of them have similarity with others, have two main divisions when the issue is the acceptance criteria applied to the plastic collapse – the main failure mechanism of VCMs – linear and nonlinear criteria. The linear acceptance criterion, governed by Tresca and von Mises stresses, as well as the elastic stress-strain relationship of the materials, is more conservative and is reflected in a heavier VCM and greater wall-thickness.

VCMs designed by this methodology used to have many problems in the field during installation and removal due to their high bending stiffness associated with the gooseneck pipe design. Although in some applications the safety factor tends to be high, the manufacturing cost for this equipment may be noncompetitive. On the other hand, the nonlinear acceptance criterion, governed by plastic strain and a stress-strain behavior established by elastic-perfectly-plastic or plastic curves of the materials, enables an effective understanding of the mechanical behavior of the VCM and how this equipment behaves before the onset of this mechanical failure. Recall that each aforementioned criterion is defined in the local and global criteria for each respective standard, as well as the functional criterion, which imposes that the equipment must not have any loss or limitations in its functions after being subjected to loading step.

In the following publications found in the literature, the VCM was considered as a secondary issue in the research work. Anyway, this equipment has been an increasing focus in the academy and industry due to many gaps and opportunities available, in the analysis and the calculation methodology. One of the great contribution in the VCM analysis – limited to its curved pipe, as known as gooseneck pipe – was covered in [¹⁹], in which the curved pipe was analyzed considering the variation of the its curvature radius and the influence of the external loads transferred from the flexible pipeline connection. The acceptance criteria used in his work was the nonlinear elastic-plastic analysis established in the ISO 13628-7. Recall that any subsequent event was investigated; considering neither the materials derating. Another relevant publication is presented in [²⁰], who were responsible for identifying the parameters that govern the VCMs during their installation in DVC.

They research concluded that the main parameter that influences the VCM during its subsea installation is the bending stiffness of the flexible pipeline. This result is really important because the production pipelines have high values of bending stiffness reflecting in elevated loads transferred to the VCMs and in the difficulty of installing this one on the permanent equipment respecting their entrance angle constraints imposed in this guide system design and hydraulic connector specifications.

The main industry standards for design and structural analysis of subsea equipment were analyzed in [²¹]. Four components were evaluated in their publication: an open-ended cylindrical vessel subjected to external pressure, a cylindrical vessel subjected to internal pressure, a load shoulder, and a straight pipe subjected to bending moment; all components are defined as thin-walled vessels. Among the failure modes indicated in the industry standards, plastic collapse, and fatigue, these can be established as primary and more common modes. Regarding the equipment subjected to static or few loading cycles, the fatigue mode can be neglected in the presence of a plastic collapse mechanism. As expected, the linear criterion indicated in the analysis standards covered, e.g., API STD 17G [¹⁶], API STD 6X [¹⁷], and ASME BPVC.VIII.2 [⁷], presented a more conservative approach compared to the limit load and elastic-plastic criteria.

2. MATERIALS AND METHODS

2.1. Finite element model

The finite element model was developed using ANSYS Workbench 2022 R2. Several steps were automated to reduce the time consumed during the pre-processing and post-processing stages. These automated steps were incorporated into the software environment by Commands tool, and the corresponding code was written in ANSYS Parametric Design Language (APDL).

2.1.1. Material

Table 2 summarizes the mechanical properties of the VCM materials used in the finite element model.

Table 2
Mechanical properties of the VCM materials.

COMPONENT	MATERIAL	SMYS	UTS	YOUNG MODULUS	POISSON RATIO
Gooseneck pipe	API 5L Grade X65	448 MPa	517 MPa	190 GPa	0.27
Lower flange	AISI 8630	517 MPa	655 MPa	206 GPa	0.30
Upper flange	AISI 8630	517 MPa	655 MPa

2.1.2. Material stress-strain curves

Stress-strain curves for the VCM materials were generated by an isotropic hardening model established by ASME BPVC.VIII.2 (Annex 3-D) [⁷] as discussed in the Section 1.3.

Figure 2 presents the stress-strain curves for the VCM materials.

Figure 2
Stress-strain curves for the VCM materials [⁷].

The plastic strain represents the local criterion for elastic-plastic behavior according to [⁸]. For the global criterion, i.e., overall structural instability of the VCM, this parameter is implicitly accounted in the total strain, which has an admissible value of 2%. This acceptance criteria will be explored in the Section 2.2.

2.1.3. Geometry

The VCM geometry was modeled using SpaceClaim 2022 R2, including only the components that withstand the external loads from the flexible pipeline being represented. For the 1^st DVC, the lower and upper flanges together with the gooseneck pipe are sufficient for providing an accurate representation of the VCM. The geometric parameters of the gooseneck pipe are defined as follows: inner diameter 139.7 mm, outer diameter 212.7 mm, curvature radius 561.0 mm, lower length of the straight-pipe 336.6 mm, upper length of the straight-pipe 111.1 mm, and the angle with vertical axis 60º. The upper and lower flanges follow standardized dimensions defined in API SPEC 6A for the 7.1/16″-10k flange type [²²]. Given the symmetry of the geometry, boundary conditions, and applied loads, only half of the VCM was modeled to reduce CPU time.

Figure 3 illustrates the VCM geometry.

Figure 3
VCM geometry.

2.1.4. Mesh

The VCM mesh was generated using SOLID186 elements from the ANSYS element library. This element featuring 20 nodes, with 3 and 60, respectively, DoFs per node and per element. This element is well-suited for accurately computing stresses and strains within the VCM. Upon completion of the mesh generation process, the VCM geometry comprised 90,902 nodes and 18,546 elements.

Figure 4 illustrates the VCM mesh generated.

Figure 4
VCM mesh.

2.1.5. Boundary and loading conditions

The first boundary condition applied in the finite element model enforces the symmetry plane on x-y local axis – see the local coordinate system triad on the upper flange – as shown in Figure 5(a).

Figure 5
Boundary conditions.

This boundary condition is equivalent to a frictionless support, in which the faces highlighted in red are allowed to slip relative to each other but cannot separate or move in any other direction. The last boundary condition is defined as a fixed support on the lower face of the lower flange, as shown in Figure 5(b). This boundary condition represents the locked state of the VCM’s hydraulic connector when engaged with the permanent equipment, e.g., VXT or Manifold.

The loading condition is characterized by the external loads transferred from the flexible pipeline to the swivel connected to the upper flange.

Regarding the swivel flange, for any applied loading condition, it is not the weakest component of the VCM. Therefore, it can be omitted on the finite element model; however, its effect is represented by a remote point located on the upper flange face.

Figure 6 presents the loading condition, including the local coordinate system on the upper flange and the remote point positioned ahead in space representing the swivel length with equal to 650.0 mm.

Figure 6
Loading conditions.

From Figure 6, the positive values of tension, shear, and bending moment are defined, respectively, as follows: Fx (x+, normal to the upper flange), Fy (y+, tangential to the upper flange), and Mz (z+, counterclockwise rotation). As discussed previously, the loads are applied to the VCM upper flange through an APDL script executed within the ANSYS environment using Commands interface. In this APDL routine, the initial load is defined as an estimative of the admissible load, and it is incremented or decremented at each iteration until the global and local acceptance criteria for elastic-plastic analysis, as specified in [⁸], are satisfied. The numerical method that governs these iterative load increments and decrements is the bisection method [²³]. This choice was made because the method guarantees convergence when the root lies within the interval [a,b], although its convergence rate is low compared with other numerical methods, e.g., Newton-Raphson [²⁴].

2.2. Acceptance criteria

2.2.1. ISO 13628-7

The ISO 13628-7 standard [⁸] is widely used in the Oil & Gas industry for Completion/Workover riser systems, and it includes a section, Annex D, dedicated to requirements for Structural Resistance Methods. The main purpose of this section is to define the requirements, assumptions, and criteria for calculating the plastic collapse and ultimate load capacity for a given mechanical system. An exception here is that fatigue and buckling are neglected due to their irrelevance as failure mechanisms for VCMs.

The methods appointed in Annex D [⁸] can be divided into three categories: analytical, numerical, and experimental. For analytical methods, the calculation is based on mathematical equations derived from fundamentals of mechanics and, in some cases, calibrated through laboratory tests. On the other hand, numerical methods can be used in calculation steps for achieve high accuracy in displacements, strains, and stresses, as well as to enable a more comprehensive understanding of the mechanical behavior. Furthermore, only the numerical methods of the Finite Element Method (FEM) and the Boundary Element Method (BEM) are permitted. Regarding full-scale experimental tests, they shall be performed to verify compliance with the standard in question. Nevertheless, if the engineer chooses an analytical approach, another method with higher accuracy, such as numerical or experimental one, must be applied to validate the design. The von Mises stress and Tresca criteria for ductile materials can be used in analysis, although the latter is slightly more conservative.

Given a numerical analysis or design-by-analysis approach, the latter of which aligns with the goal of this work, the following alternatives are available:

1. Elastic analysis: based on an elastic material;

2. Limit analysis: based on an elastic-perfectly-plastic material and small-deformation theory;

3. Plastic collapse analysis: based on strain hardening of the material and large-deformation theory. The stress-strain model herein assumes isotropic hardening.

The plastic collapse analysis will be explored in detail, as this method, when coupled with a finite element model, accurately represents the phenomenology of interest – namely, the expansion and contraction of the yield surface of the VCM after an idealized-discrete loading, unloading, and subsequent loading step.

2.2.2. Ultimate strength limit

As described in Section D.1.3 [⁸], the design load effect shall satisfy the following criterion:

(20) $$S_d\le F_d{R}_{uc}$$

where F_d and R_uc are, respectively, the design factor and the plastic collapse or ultimate capacity.

Equation (20) can also be expressed in terms of Equation (21).

(21) $$S_d\le \frac{2}{\text{3}}{C}_f{R}_{uc}$$

where C_f is the design condition factor.

Table 3 presents the design condition factor for each load condition scenario and its related failure mode. In this study, the load condition was established as normal, with a design condition factor equal to 1.00.

Table 3
Design condition factor (Table D.1 [⁸]).

LOAD CONDITION	FACTOR	FAILURE MODE
Assembly and disassembly	1.35	Based on actual design values at assembly/disassembly temperature
Mill/FAT hydrostatic pressure test	1.35	Based on actual design values at test temperature, fluid (hydrostatic)
Normal operation	1.00	Based on corroded wall thickness at design metal temperature
Extreme operation	1.20	Based on corroded wall thickness at design metal temperature
System pressure test	1.20	Based on corroded wall thickness at the test temperature
Temporary operation	1.20	Based on corroded wall thickness at actual metal temperature
Accidental operation	1.50	Based on corroded wall thickness at actual metal temperature

2.2.3. Plastic collapse premises

In the plastic collapse analysis, all relevant components that contribute to the VCM strength must be modeled in the FEA; in general, these components are included in the loading path of the system being evaluated. In this context, including full VCM components such as anodes, Remotely Operated Vehicle (ROV) panel, shackle, and adjacent structures are not relevant for FEA purposes.

The limit-load or plastic-load analysis must satisfy the following requirements:

• Global criterion

  • The load that causes overall structural instability, i.e., the maximum load achieved in the FEA for which full convergence is guaranteed, must be considered. The principal structural strain limit must be less than or equal to 2%.

• Local criterion

  • The load that produces the equivalent von Mises plastic strain must not exceed the value given by Equation (22) at any point of the VCM components.

(22) $${\epsilon }_{peq}\le min\left[0.1;0.5\left(1-\frac{{\sigma }_y}{{\sigma }_u}\right)\right]$$

where ε_peq, σ_y, and σ_u, are the equivalent plastic strain, the design yield strength, and the design ultimate tensile strength at design temperature, respectively.

• Functionality criterion

  • This criterion is responsible for establishing an additional constraint regarding the functional utilization of the VCM. In fact, the global and local criteria must not cause any damage to the VCM that could result in the loss of its full functionality under normal to extreme scenarios. For accidental scenarios, although the loss of several functionalities is not expected, such losses may occur; however, they must not lead to a catastrophic failure. Therefore, after the VCM has been subjected to this last scenario, it is strongly recommended to perform its maintenance.

3. RESULTS AND DISCUSSION

3.1. Admissible loads

Table 4 presents the parameters calculated for each VCM material, considering the multilinear isotropic hardening model presented in Annex 3-D [⁷] and explained in Section 1.3. The values of H, γ₁, γ₂, and ε_t were extracted at yield stress point, just for reference. The parameters presented in Table 3 are used to calculated the stress-strain curves of the VCM materials employed in the finite element model, as show in Figure 2.

Table 4
Multilinear isotropic hardening parameters (Annex 3-D [⁷]).

PARAMETER	EQUATION	MATERIAL
		API 5L GRADE X65	AISI 8630
εt	${\epsilon }_t=\frac{{\sigma }_t}{E_y}+{\gamma }_1+{\gamma }_2$	4.29.10-3	4.53.10-3
γ 1	${\gamma }_1=\frac{{\epsilon }_1}{2}\left[1.0-\tanh \left(H\right)\right]$	1.84.10-3	1.89.10-3
γ 2	${\gamma }_2=\frac{{\epsilon }_2}{2}\left[1.0+\tanh \left(H\right)\right]$	8.85.10-5	1.29.10-4
ε 1	${\epsilon }_1={\left(\frac{{\sigma }_t}{A_1}\right)}^{\frac{1}{m_1}}$	1.88.10-3	1.92.10-3
A 1	$A_1=\frac{{\sigma }_{ys}\left(1+{\epsilon }_{ys}\right)}{{\left[\ln \left(1+{\epsilon }_{ys}\right)\right]}^{m_1}}$	546.21 MPa	714.83 MPa
m 1	$m_1=\frac{\ln \left(R\right)+\left({\epsilon }_p-{\epsilon }_{ys}\right)}{\ln \left[\frac{\ln \left(1+{\epsilon }_p\right)}{\ln \left(1+{\epsilon }_{ys}\right)}\right]}$	3.15.10-2	5.18.10-2
ε 2	${\epsilon }_2={\left(\frac{{\sigma }_t}{A_2}\right)}^{\frac{1}{m_2}}$	4.92.10-3	7.15.10-3
A 2	$A_2=\frac{{\sigma }_{uts}\exp \left(m_2\right)}{m_2^{m_2}}$	685.61 MPa	965.16 MPa
H	$H=\frac{2\left\{{\sigma }_t-\left[{\sigma }_{ys}+K\left({\sigma }_{uts}-{\sigma }_{ys}\right)\right]\right\}}{K\left({\sigma }_{uts}-{\sigma }_{ys}\right)}$	-2.00	-2.00
R	$R=\frac{{\sigma }_{ys}}{{\sigma }_{uts}}$	0.87	0.79
εys	${\epsilon }_{ys}=0.002$	0.002	0.002
K	$K=1.5R^{1.5}-0.5R^{2.5}-R^{3.5}$	0.25	0.34
σuts,t	${\sigma }_{uts,t}={\sigma }_{uts}\exp \left(m_2\right)$	560.18 MPa	743.19 MPa

Table 5 presents the summary of admissible loads calculated for step 1 – tension, shear, and bending moment. It is worth recalling that this step is characterized by the absence of previous plastic strain, i.e., zero. Recall that “positive” refers to maximum and “negative” refers to the minimum value.

Table 5
Positive and negative admissible loads for step 1.

TENSION		SHEAR		BENDING MOMENT
NEGATIVE	POSITIVE	NEGATIVE	POSITIVE	NEGATIVE	POSITIVE
-222.89 kN	228.94 kN	-162.40 kN	162.00 kN	-266.82 kN.m	270.18 kN.m

From Table 5, for tension and bending moment loads, we can observe that the positive magnitudes, in absolute terms, are always slightly greater than the negative ones. Moreover, the relative difference between the negative and positive load values is more significant when compared with the shear. In fact, this difference follows the sequence: tension > bending moment. For the shear load, only the case that case exhibited the opposite behavior, i.e., the negative magnitude was greater than the positive magnitude. Nevertheless, considering the “tolerance” in the APDL script for the utilization factor within [99%; 100%] to achieve the ultimate load, these values could technically exhibit the same behavior as the others loads. Finally, the results suggest an almost symmetric structural behavior due to the proximity between the negative and positive values obtained from the FEA. The admissible loads indicated above are compatible with the flange 7.1/16″–10k structural capacity described in API TR 6AF [²⁵]. However, as expected and established as a premise in the VCMs design, the weakest point for any load must be the gooseneck pipe.

Figure 7 shows the total strain results for the positive (a) and negative (b) tension in step 1.

Figure 7
Positive and negative tension in step 1.

Figure 8 shows the total strain results for the positive (a) and negative (b) shear in step 1.

Figure 8
Positive and negative shear in step 1.

Figure 9 shows the total strain results for the positive (a) and negative (b) bending moment in step 1.

Figure 9
Positive and negative bending moment in step 1.

From Figures 7–9, we can observe the difference between the failure mechanisms promoted by each load application: positive tension, negative tension, positive shear, negative shear, positive bending moment, and negative bending moment, respectively. For tension load, the total strain is highly concentrated at the beginning of the gooseneck pipe, close to the lower VCM flange – a classical plastic hinge mechanism resulting from plastic collapse of the overall cross-section. For the shear load, a different total strain gradient is observed, and the most concentrated region located along the centerline of the curvature radius of the gooseneck pipe. Finally, for the bending moment load, the entire gooseneck pipe is subjected to a well-distributed total strain near the upper flange. For any loading condition evaluated, the most critical region is the intrados of the gooseneck pipe rather than its extrados. Thus, considering the loads discussed above, a critical structural instability is expected to be under tension due to the concentrated failure behavior associated with localized plasticity involved; where tension represents the load condition in which the total strain extends across the entire cross-section of the gooseneck pipe.

3.2. Sensitivity analysis

3.2.1. Tension

Figure 10 shows the results of positive (a) and negative (b) admissible tension for step 1 and step 2. Recall that step 1 uses the original geometry as its initial state, i.e., the undeformed geometry without plastic strain, whereas step 2 uses the strain state obtained in step 1 as its initial condition. For instance, if a plastic strain of 0.50% is inferred in step 1, then in step 2 the positive achievable total strain, according to the global criterion, will be 1.50%. This value results from subtracting the admissible total strain of 2%, applied to both steps, from the plastic strain accumulated in step 1 (0.50%). The blue and gray curves represent step 1 and step 2 from the FEA simulations, respectively. These results are presented as a dimensionless¹ parameter, using a plastic strain increment of 0.05% from step 1 for each load case.

Figure 10
Positive and negative admissible tension versus total strain.

For positive tension, the admissible value calculated in step 2 decreases linearly until it reaches the last load case, in which total strain is 2%. This behavior occurs because the VCM already exhibits initial damage², which reduces its structural capacity for that load. However, in some cases, the isotropic hardening of the materials can lead to an increase in the VCM’s structural strength due to cold hardening effects. If the gray curve lies below the blue curve, it indicates that the VCM’s structural strength has been reduced. For example, considering that the plastic strains in step 1 and step 2 are 1.95% and 0.05%, respectively, the current positive admissible tension is reduced by almost 15%. This value results from the difference between 1.00 and 0.85.

Thus, the remaining structural capacity of the VCM must be evaluated using the gray curve from step 2 rather than step 1, as is commonly done in some industrial applications for pressure vessels based on elastic-plastic analysis criteria [⁸]. For minimum tension, up to a certain level of plastic strain in step 1, the VCM improves its structural strength due to the isotropic hardening phenomenon shown in the stress-strain curves. So, in this case, the VCM shows an increase in structural capacity.

Figure 11 shows the resultant tension, in Newton (N), versus total strain. This parameter corresponds to the square root of the sum of squares of the positive or negative tension values in step 1 and step 2, as shown in Equation (23).

Figure 11
Resultant tension versus total strain.

(23) $$T_{r\pm =}\sqrt{T_{1\pm }{}^2+T_{2\pm }{}^2}$$

where T_r±, T₁±, and T₂± are, respectively, the positive or negative resultant tension for both steps, the positive or negative admissible tension in step 1, and the positive or negative admissible tension in step 2. This equation, with minor modifications, will later be applied to shear and bending moment loads.

From Figure 11, a nonlinear response is observed up to total strain of 1.00% in step 1. Beyond this total strain, the resultant positive tension becomes constant. The curve of resultant positive tension increases for total strain values below 1.00% in step 1, and no gross plastic strain is observed in the gooseneck pipe within this range. However, beyond this point, the overall structural instability develops in the cross-section due to the formation of a plastic hinge [²⁶, ²⁷]. These results suggest that the VCM’s strength is more sensitive to negative tension (compressive) than to positive tension (tensile). Although, for a certain level of plastic strain in the previous step, the structural capacity of the equipment may improve.

3.2.2. Shear

Figure 12 presents the positive (a) and negative (b) admissible shear for step 1 and step 2. This result denotes the same rule as for tension described before: a growth rate of 0.05% of plastic strain over the interval [0; 2%].

Figure 12
Positive and negative admissible shear versus total strain.

From Figure 12, it is observed that positive and negative shear exhibit different behavior in step 1. For negative shear, a minor increase in VCM strength is observed within the interval [0.25%; 1.00%]. Considering step 2, the results suggest less influence of negative shear compared to positive shear. Thus, negative values of tension and shear have less influence on VCM structural strength than positive values.

Figure 13 shows the resultant shear versus total strain. This parameter is described in the Equation (24).

Figure 13
Resultant shear versus total strain.

(24) $$S_{r\pm =}\sqrt{S_{1\pm }{}^2+S_{2\pm }{}^2}$$

where S_r±, S₁±, and S₂± are, respectively, the positive or negative resultant shear for both steps, the positive or negative admissible shear in step 1, and the positive or negative admissible shear in step 2.

From Figure 13, both curves indicate equivalent behavior, but they lie remarkably similar to each other compared to the tension discussed earlier. These results suggest that the VCM structural strength has almost the same for positive and negative shear.

3.2.3. Bending Moment

Figure 14 presents the admissible positive (a) and negative (b) bending moment versus total strain. Similar behavior is observed in step 1 when compared with tension and shear results discussed in the previous sections. For step 2 and up to 0.75% plastic strain in step 1, the blue curve does not change significantly.

Figure 14
Positive and negative admissible bending moment versus total strain.

Equation (25) presents the positive resultant admissible bending moment.

(25) $$B{M}_{r\pm =}\sqrt{B{M}_{1\pm }{}^2+B{M}_{2\pm }{}^2}$$

where BM_r±, BM₁±, and BM₂± are, respectively, the positive or negative resultant bending moment for both steps, the positive or negative admissible bending moment in step 1, and the positive or negative admissible bending moment in step 2.

From Figure 15, a nonlinear behavior is observed up to a total strain of 0.75% in step 1. The reason for this is the absence of gross plastic strain and, consequently, the lack of a plastic hinge in the gooseneck pipe for total strain levels below 0.75% in step 1. Both curves indicate equivalent behavior and are identical. These results suggest that positive and negative resultant bending moment exert the same influence on the VCM’s structural strength.

Figure 15
Resultant bending moment versus total strain.

4. CONCLUSIONS

In this work, a sensitivity analysis was performed considering the application of external loads through a flexible pipeline on the upper flange of an illustrative VCM. Thus, tension, shear, and bending moment were applied individually to the VCM to verify the influence of each external load on the structural strength, using an isotropic hardening model to develop the stress-strain relationship of the materials. The positive and negative directions, i.e., the maximum and minimum values, respectively, were investigated across the overall plastic strain levels inferred in step 1. From the FEA results, it was observed that negative tension has a greater influence on the downgrade of the VCM structural capacity when compared with its positive direction, i.e., tensile. Moreover, the resultant tension exhibited a greater structural asymmetry for the VCM when compared its results with shear and bending moment loads. The resultant shear and resultant bending moment have shown structural symmetric for the positive and negative directions; the latter is still closer. For instance, VCM has practically the same structural response when considering the same magnitude for positive or negative values.

Despite that, focusing only on the loading magnitude, the VCM exhibited critical behavior when the pure shear load was applied. Another relevant fact was observed due to the isotropic hardening model used to develop the stress-strain curves of the VCM materials: by increasing the plastic strain level inferred in step 1, the VCM strength was reduced in the subsequent loading cycle, i.e., step 2 [²⁸]. Thereby, an exception to this behavior was only observed for small plastic strains under negative tension and negative shear loads. The isotropic hardening model from ASME BPVC.VIII.2 (Annex 3-D) enables a finer understanding of the influence of tension, shear, and bending moment on the VCM structural capacity, in which representatively describe the total strain distribution across the VCM behavior, and its mechanical phenomenology involved in the derating and the upgrade of the structural capacity of the equipment in question [²⁹]. Finally, these conclusions help to achieve greater design robustness and constraints for the VCMs and, consequently, to reach a safer operational scenario in the Oil & Gas industry.

5. DATA AVAILABILITY

The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.

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• 1
  Normalization process; division of the load and maximum load magnitude as an absolute value.
• 2
  Plastic strain or plasticity.

Publication Dates

• Publication in this collection
  06 July 2026
• Date of issue
  2026

History

• Received
  20 Dec 2025
• Accepted
  20 May 2026