The Behavior of Elbow Elements at Pure Bending Applications Compared to Beam and Shell Element Models

Subsea pipelines are increasingly being required to operate under high pressures and temperatures. As such, the pipeline tends to relieve the resulting high axial stress in the pipe wall by lateral buckling or upheaval buckling [1–4]. The most common failure modes within lateral or vertical buckling are local buckling, fatigue, and fracture [1,5,6]. The pipeline can also experience large bending formation due to lateral displacement resulting from the on-bottom current and wave effect, third-party interference from trawl fishing activities, etc. Excessive local buckling can cause significant ovality to the pipe cross section [7,8]. In the event the ovality of the pipeline is increased to a certain level, the pipeline maintenance through pigging operations can be compromised, directly affecting the serviceability limit state of the pipeline [6].

Furthermore, subsea pipelines are often subjected to bending in reverse directions during installation. The pipe experiences an overbend on the installation barge and the attached stinger. Once the pipeline leaves the installation barge and stinger, it begins to change the bending direction, known as sagbend, until it reaches its highest value near the seabed touchdown. When the pipeline is laid on an uneven seabed, it tends to bend to conform to the bottom seabed profile. Other factors, such as trenching and crossing, can also cause pipeline bending during installation. For bending within the elastic region, the ratio of the bending moment to the curvature is known as flexural rigidity. As the curvature increases and the points furthest from the neutral axis begin to yield, the relationship between the bending moment and the curvature changes. The curvature significantly increases with a slight increase in the bending moment, influenced by strain-hardening (which tends to reduce the strain) and ovality (which tends to increase it).

Given these challenges, the mechanical response of pipelines under bending conditions is critical to ensuring their structural integrity, especially in deepwater installations. This study investigates the response of ELBOW31 and ELBOW31B element types under pure bending conditions, using shell and beam element models for benchmarking. The effectiveness of these elements is assessed through detailed comparisons with these established models.

The study focuses on pure bending as it serves as a fundamental baseline for understanding more complex loading scenarios in pipeline systems. Pure bending allows for the isolation of bending effects, providing a clear reference point against which the influence of additional factors, such as internal or external pressures, can be assessed. By first understanding how pipelines behave under pure bending, the study sets the stage for future research into more complex conditions.

The elbow element is particularly significant in finite element analysis (FEA) of pipelines, especially when addressing the unique challenges posed by curved sections within long-distance transmission systems. Modeling elbow elements introduces complexities due to localized phenomena such as ovalization, where the pipe's cross section deforms from circular to elliptical, and warping, which involves twisting of the cross section. These deformations lead to stress intensification, particularly under high internal pressures or when the pipeline is subjected to thermal expansion and contraction. The elbow element is specifically designed to capture these effects, providing a realistic representation of the stresses and strains that occur in curved pipeline sections, which are often the most vulnerable parts of the pipeline. This precision is essential for ensuring the integrity and reliability of pipelines under various operational conditions.

For the elbow element models, analyses focus on factors that significantly affect the behavior of the elbow elements, such as the number of ovality modes and the application of the NOWARP and NOOVAL conditions to the end nodes [9]. Additionally, the study investigates the validity of using a single (1D length) element model by comparing its results with those from ten-element finite element (FE) models, as well as longer element length models. The article also introduces a comparison of the bending moment–strain curve when using different values for Poisson's ratio, while the context focuses solely on steel pipelines.

The main body of the study focuses on a pipe section with a diameter to wall thickness (D/t) ratio of 34, a common configuration in pipeline design. The pipe has an outer diameter (D) of 323.85 mm and a wall thickness (t) of 9.52 mm of X65 carbon steel, dimensions typically used in deepwater pipelines to balance strength and flexibility. Various model lengths are evaluated to determine their impact on the bending moment–strain relationship. The results indicate that a model length equal to six pipe diameters exhibits a hardening effect when the total strain exceeds 3.5%, although a strain up to 1% is considered sufficient for pipeline design. These findings are crucial for accurately predicting pipeline behavior under operational conditions.

In addition to the main pipe section, two more pipe sections with different D/t ratios—D/t = 20 and D/t = 50—are analyzed in  Appendix A and  B, respectively. The pipe section with a D/t ratio of 20 has an outer diameter of 323.85 mm and a wall thickness of 15.8 mm, while the section with a D/t ratio of 50 has an outer diameter of 323.85 mm and a wall thickness of 6.5 mm. The analysis of these sections provides critical insights into how wall thickness influences the pipeline's flexibility and ovalization, which are essential for designing robust pipelines capable of withstanding bending loads.

Additionally, the study highlights that different Poisson's ratios and model lengths have minimal impact on the bending moment–strain response when no restrictions are applied. These insights are directly applicable to the design of pipelines, particularly in scenarios where bending-induced deformations pose significant risks, such as in deepwater or high-pressure environments. This study focuses on pure bending as a baseline scenario to isolate the effects of bending on pipeline integrity. Understanding the pipeline's behavior under pure bending is essential for developing accurate models that can be extended to more complex scenarios involving internal and external pressure.

The findings gained from this research contribute to the development of more reliable and robust pipeline systems, ensuring their integrity under various loading conditions. The study also suggests avenues for future research, including the combined effects of bending and internal pressure, as well as the influence of different material properties on pipeline response.

While the beam elements are widely used in piping stress analysis, the cross section of the beam does not deform and, therefore, does not properly represent the pipe behavior in the vicinity of significant curvature. Sobel [10] presented an FE analysis of the in-plane bending behavior of elastic elbows, addressing the basic physical behavior of elbows and identifying significant response characteristics. Sobel introduced guidelines for the systematic selection of the FE mesh, results interpretation, and convergence behavior, and compared results obtained from various methods of analysis. The study concluded that there was good agreement between MARC [11], the ELBOW computer model [12], and the analytical approach by Clark and Reissner [13,14].

In comparing the beam element with the elbow element, the work by Elgazzar et al. [15] on the strain concentration factor for offshore concrete-coated pipelines provides valuable insights into the implications of varying stiffness and the modeling complexities associated with these structural elements. The detailed analysis presented in their literature review underscores the importance of considering local geometries and material variations, which are critical when assessing stress and strain concentrations in different pipeline configurations.

Lazzeri [16] introduced an elasto-plastic elbow element for inclusion in the FE code library, based on Vlasov's thin wall theory [17,18]. The study demonstrated good agreement between results obtained using PAMEL (plastic analysis membrane elbows) and Ohtsubo and Watanabe's experimental and theoretical data [19]. The results were also found comparable to those obtained by MARC [11], as presented by Sobel [10].

In 1980, Bath and Almeida [20] formulated a displacement-based pipe bend element that generalizes Von Kamran's pipe radial displacement patterns [21], incorporating the ovality effect. Although the introduced element could not fully capture the three-dimensional shell behavior of elbows, it showed comparable displacements and stresses for various pipe sizes compared to previously published computational tools, with a less expensive solution. In 1982 [22], the authors reintroduced the elbow element with enhancements to capture the effects of interactions between elbows and rigid flanges, elbows with different curvatures, and elbows joining straight pipe sections. While the results showed applicable solutions, the formulation relied on several assumptions, leading to the need for further investigation into the model's validity limits.

Following these publications, Bath and Almeida identified the necessity of including the internal pressure effect on the elbow element, presenting their results in a brief note [23]. The study aimed to investigate the effect of internal pressure on the stiffness of the elbow, particularly in thin pipes. The results were generally acceptable compared to experimental data [24,25], except at higher pressures, where the neglect of midsurface strains in the ξ direction may have impacted accuracy as per [20].

Mackenzie [26] proposed three different elbow elements using user-defined elements in ANSYS [27], investigating their performance under various loading conditions. These elements, which are two-node displacement-based finite elements, incorporated axial, bending, and torsional deformation modes based on beam theory, and elbow ovalization modes based on a reduced two-dimensional shell theory. The study concluded that the PB1 elbow element showed comparable results to more complex and expensive analyses, at acceptable computational costs.

In 1990, Abo-Elkhier [28] formulated a simple yet effective elbow element following Bath and Almeida's concept [20]. This elbow element was enhanced to capture the general shell behavior of elbows and was implemented in the ELBOW computer program. The proposed elbow element was validated against other numerical and experimental results, showing acceptable performance.

Abdulhameed [29] conducted an extensive numerical investigation on pipe bends subjected to internal pressure, focusing on the “Bourdon effect,” where internal pressure tends to open the bend, generating additional hoop and longitudinal stresses. This phenomenon was found to be dependent on internal pressure, pipe radius, and bend angle, with Abdulhameed indicating that the Bourdon effect could increase stresses on pipe bends by up to 48% compared to straight pipes.

Previous studies have laid the groundwork for understanding the mechanical behavior of pipelines under various loading conditions. This study builds on these foundations by specifically addressing the limitations of the existing modeling techniques, particularly the challenges posed by elbow elements under pure bending conditions.

To date, these foundational studies have established a robust framework for understanding the behavior of elbow elements under various loading conditions. However, the increasing complexity and demands of modern subsea pipelines necessitate further refinement and validation of these models, particularly concerning their computational efficiency and accuracy under extreme conditions. This study builds on this body of work by evaluating the performance of ELBOW31 and ELBOW31B elements under pure bending conditions, with a focus on capturing ovalization and warping effects, as well as comparing these results with both traditional and modern FE techniques.

The elbow element is capable of modeling the ovalization and warping, which are localized phenomena associated with curved sections. Ovalization refers to the deformation of the pipe's cross section from circular to elliptical, while warping involves the twisting of the pipe's cross section. These deformations can lead to stress intensification, particularly under high internal pressures or when the pipeline is subjected to thermal expansion and contraction. The elbow element incorporates these effects, providing a realistic representation of the stresses and strains that occur in curved pipeline sections, which are often the most vulnerable parts of the pipeline.

This study focuses on addressing the behavior of the elbow elements subjected exclusively to pure bending moments, without the influence of internal or external pressures. By isolating the bending effects, the study aims to establish a comprehensive understanding of the elbow element's behavior under bending loads, which serves as a critical baseline for future work. In future studies, this foundational understanding can be expanded to include the combined effects of bending and pressure, whether internal or external, offering a more complete picture of the pipeline's structural response.

The insights gained from this pure bending study will serve as a foundation for more complex investigations, enabling a deeper understanding of how pressure modifies the structural response and potential failure modes. These findings are particularly important for subsea pipelines, where bending moments are common during installation and operation. Additionally, results of the nonlinear pipe stress analysis programs can still be validated by comparing them to the pure bending results provided in this paper. This helps ensure that these tools are reliable for specific applications, even with the simplified modeling approach.

This study also considers the effects of ovality modes and boundary conditions, such as NOWARP and NOOVAL, to evaluate their impact on the elbow element's performance under pure bending. This analysis provides a more detailed understanding of how these factors influence the accuracy of the elbow element models, contributing to the overall reliability of pipeline design under bending conditions.

This section presents a comparative analysis of the different element types used in this study, highlighting their applications, key characteristics, and limitations. This comparison (refer to Table 1) is essential for understanding the suitability of each element type for specific aspects of pipeline modeling and analysis.

The elbow element is particularly significant in FEA of pipelines, especially when addressing the unique challenges posed by curved sections within long-distance transmission systems. Modeling elbow elements introduces complexities due to localized phenomena such as ovalization, where the pipe's cross section deforms from circular to elliptical, and warping, which involves twisting of the cross section. These deformations lead to stress intensification, particularly under high internal pressures or when the pipeline is subjected to thermal expansion and contraction. The elbow element is specifically designed to capture these effects, providing a realistic representation of the stresses and strains that occur in curved pipeline sections, which are often the most vulnerable parts of the pipeline. This precision is essential for ensuring the integrity and reliability of pipelines under various operational conditions.

The comparison illustrates the strengths and limitations of each element type, underscoring the importance of selecting the appropriate element for different aspects of pipeline analysis. This study leverages the unique capabilities of elbow and shell elements to provide a comprehensive evaluation of pipeline behavior under pure bending conditions, while also considering the computational efficiency required for practical applications.

The finite element software ABAQUS [9] is used to model the behavior of the pipeline under pure bending loads. The model uses a four-node reduced integration shell element (S4R) to present the pipe under pure bending. The pipe is thoroughly meshed along the pipe length; a finer mesh is used in the middle of the pipe to accurately capture the failure mechanism during bending. The shell model employs nine integration points through the wall, the reduced integration shell elements are employed to accurately capture the bending behavior with particular attention to boundary conditions and the application of loads that simulate realistic pipeline operational scenarios.

The pipe steel material is modeled to capture the pipeline behavior in both elastic and plastic ranges. The pipe joint is modeled to simulate pinned boundary conditions on the right end, all degrees-of-freedom are restricted except rotation around the Z-axis. The left end of the model simulates roller boundary condition, displacement along the pipe X-axis as well as rotation around the Z-axis are unrestrained; all other displacements and rotations are restricted. Figures 1 and 2 show a schematic diagram of the pipe model and the FEA model, respectively.

Rotation around the z-axis is applied at both ends through two steps: initially, 0.015-rad rotation is applied using a static step, resulting in longitudinal strains on the pipe within the elastic range (<0.2%). Subsequent rotation is applied using the static RIKS method, straining the pipe above the elastic limit.

Beam-element models are developed in ABAQUS [9]. The models are created with Pipe-element PIPE31, Elbow-element ELBOW31, and Elbow-element ELBOW31B. PIPE31 is a 2-node (single element) linear beam pipe-element in space following the Timoshenko (shear flexible) beam theory [30]. In this work, a thin-walled cross section model is used with PIPE31. On the other hand, ELBOW31 and ELBOW31B are two-node (single element) beam element types that consider the nonlinearity response of the circular cross section pipes under bending loads due to ovalization and warping effects, that are made continuous from one element to the next, known as the “Brazier effect” [31]. While ELBOW31 is the complete elbow element type, a simplified elbow element type ELBOW31B is used, which considers only the effect of the ovalization, while the warping and the axial gradient of the ovalization are neglected.

The choice of boundary conditions, such as NOWARP and NOOVAL, is critical in the elbow element models (ELBOW31 and ELBOW31B). These conditions impact the accuracy of the simulation, particularly in representing the complex stress states in curved sections of the pipeline.

In this study, both Elbow and Pipe element types are used for comparison purposes. Single and ten (10) element model lengths are investigated, while one diameter (1D)-element length is maintained for the models. At the right end of the model, the boundary conditions are set to constrain the displacement along the z-axis, the rotation around the X-axis, and the rotation around the Y-axis, while the left end is set to be fully constrained. The bending moment is applied by applying rotation around the Z-axis to the model's right end. A schematic diagram of the beam element model is shown in Fig. 3. It is worth noting that for the beam elements model, the boundary conditions and the bending moment can be applied directly on the node, while for shell element models, boundary conditions and loads are applied to a reference node connected to the pipe end via kinematic constraints.

Analysis has captured material behavior within and beyond the elastic range to a total strain of 6%; a target total strain level of 1% is found sufficient to analyze the pipe behavior for most cases of loading, however, results are presented for larger strains. The stress–strain curve of the steel material used in the analysis is shown in Fig. 4.

Understanding the behavior of the pipelines under pure bending is essential for establishing a baseline for more complex loading scenarios such as those involving internal and external pressures. This baseline allows for more accurate predictions of the pipeline performance and the identification of the potential failure modes.

The shell element employed in this study uses four nodes reduced integration (S4R) in ABAQUS [9]. The bending moment–strain results for various model lengths with the same d/t ratio (diameter to wall thickness) of 33 are presented in Fig. 5. Results show that changing the length has minimal effect on the pipe response until 3.5% strain. The 6D-length model presents a stiffening effect beyond 3.5% strain. For all cases, the pipe is subjected to pure bending moment; hence, internal pressure is set to zero.

Figure 6 presents the bending moment versus strain curves obtained for the ELBOW31-single element model, with different Fourier ovality mode numbers, the figure shows the effect of changing the ovality modes only while not applying wrapping or ovality restrictions to pipe nodes. The effect of imposing the NOWARP boundary condition (i.e., restraining the warping deformation) at both ends has also been investigated and presented in Fig. 7. For both figures, the bending moment–strain curves for the Pipe and Shell element models are shown for comparison and verification.

It should be noted that the beam-element model length is equal to one diameter of a 12.75-inch pipe (323.85 mm); for a symmetrical model, the beam model is set to be equivalent to half length of the shell element model. The end conditions of the Shell model are controlled by distributed couplings, to ensure not restricting the ovality, this configuration is equivalent to the absence of the NOOVAL boundary condition for the ELBOW31 element. For all cases, the Poisson ratio is set to zero.

Figures 6 and 7 show that utilizing one (1) ovality mode within the ELBOW31 element model produces the best match when compared to the Pipe element model. When the number of ovality modes increases, a softening effect is noticed when NOWARP is not applied, and the pipe experiences lower capacity than both the Pipe and the Shell element models. The elastic capacity of the elbow model is considerably reduced when the number of ovality mode exceeds two (i.e., 3–6 ovality modes). For instance, the ELBOW31 element model with 3 ovality modes shows a reduction of 45 kN.m in the elastic capacity when compared to the Pipe and the Shell element models; hence, it is believed that the results of the ELBOW31 element models with ovality modes exceeding two (2) are not correctly representing the pipe behavior under pure bending.

On the other hand, Fig. 7 shows that when the NOWARP condition is applied at the model ends, the agreement is seen to be remarkably good compared to the Pipe element model for the first ovality mode. For ovality modes equal to or more than two, the agreement was sustained up to 2.0% strain when compared to the Shell element model. That is coherent with the ABAQUS User's Manual [9], where it is stated that “Although the elbow elements appear as beams, they are, in fact, shells, with quite complex deformation patterns allowed.”

When zero ovality mode is used, results show a significant increase in the elastic capacity for both ELBOW31 and ELBOW31-NOWARP, with a larger bending moment required to reach the elasticity limit of the pipe, over 64 kN.m for both the Shell and Pipe element models. It should be noted that these results contradict the ABAQUS User's Manual, which states that “when the ELBOW31 element is used with zero ovality mode and the Poisson's ratio is equal to zero, the model should become a simple Pipe element.” Therefore, results obtained from the elbow elements indicate that zero ovality mode is not able to capture the correct bending moment–strain response.

Figure 7 shows that the softening behavior of the bending moment–strain curve obtained from the Shell element model is found to be in good agreement with the ELBOW31 element with the NOWARP boundary condition, specifically with four or greater ovality modes. While those cases show that the bending moment begins to drop at an earlier strain (approximately 2.0%) compared to the Shell model, which shows that the bending moment begins to drop at 3% strain; however, it should be noted that the desirable strain during the pipeline design does not exceed 1% in majority of the cases. The variance in pipeline behavior is noted for extremely high strain levels (2% and above), which is not aimed for pipeline design purposes, however, bending moment-versus-strain response for high strains is shown in the figures for illustrations only.

The behavior of the ELBOW31 elements considering NOOVAL boundary conditions at both ends is investigated; however, Fig. 8 shows unrealistic results during both elastic and plastic loading; hence, the study does not recommend utilizing elbow elements with NOOVAL boundary conditions. Further investigation is recommended to investigate the applicability of the NOOVAL boundary conditions to ELBOW31 elements; however, it is not within the scope of this study.

The bending moment versus strain curves resulting from applying both the NOWARP and NOOVAL boundary conditions to the ELBOW31 element are investigated, for all cases the Poisson ratio is set to zero. Figure 9 shows the relationship considering various ovality modes. When both restrictions (NOWARP and NOOVAL) are applied, the ELBOW31 element produces bending moments higher than what is produced from both the Shell and the Pipe elements. The effect of applying both the NOWARP and NOOVAL boundary conditions is found to be equivalent to using zero ovality modes as indicated in Figs. 6 and 7, the resulting bending moment versus strain is found to be similar regardless of the utilized ovality modes. Thus, the resulting bending moment–strain relationship is not considered reliable; further investigation is recommended, however, it is not part of the scope of this study.

The ELBOW31B element is a simpler model of the elbow-element type where the axial gradient of ovality is not considered, and therefore it is not transferred to the adjacent elements. This condition eliminates the requirement for the application of NOOVAL boundary conditions; hence ELBOW31B element ends are free to ovalize.

Figure 10 presents the bending moment versus strain relationship for the ELBOW31B model assuming various ovality modes.

For cases of one (1) to six (6) ovality modes, results of the ELBOW31B element model are found to be in good agreement with the Shell and Pipe elements for total strain values up to 1.0%. Similar to what is observed for the ELBOW31 element models, results of one ovality mode are found to be in good agreement with the Pipe element. On the other hand, the zero ovality mode produces an overestimated result.

Ovality mode numbers equal to or greater than two (2) produce bending moment–strain curves in very good agreement with the Shell element model up to 2.0% strain. Results show no remarkable difference between ovality modes two and three when used. On the other hand, for models with four or greater ovality modes, the similarity is sustained only up to 4.0% strain; beyond that, both modes with four and five ovality modes maintain the similarity, while the model with six ovality modes produces a significantly softer behavior.

As observed with the ELBOW31 element model when the NOWARP boundary condition is applied, the ELBOW31B with four or greater ovality modes produced softening effects on the bending moment–strain curves. Models with four or five ovality modes presented the closest results to the curves from the Shell element case. For all cases, the Poisson ratio is set to zero.

Figure 11 shows a comparison between ELBOW31 with NOWARP conditions and ELBOW31B. It can be observed that results show an agreement when zero to three ovality modes are used. For larger ovality modes, the agreement is obtained only up to 3.0% total strain. For total strains between 3.0% and 5.0%, the bending moment resulting from the ELBOW31B element is found to be slightly higher than what is produced from the ELBOW31-NOWARP. At higher strains, the results return to show very good agreement again.

The effective Poisson's ratio is a property of the cross section, which computes the cross section area change due to normal or axial deformation. This effect takes place when a beam element is subjected to a considerably significant axial stretch, causing a reduction or an increase in the cross section area, in ABAQUS, the effect of Poisson's ratio is not considered in elbow elements [9].

This section presents the results of the bending moment–strain relationship for both the elbow and pipe element models, when two different values of Poisson's ratio are used, i.e., 0.0 and 0.5. For illustration, zero Poisson's ratio reflects no cross section change with axial stretch; while 0.5 Poisson's ratio implies that the overall response of the section is incompressible. The comparison utilizes ELBOW31, ELBOW31B, and PIPE31 element models.

Figure 12 below shows that, while the effective Poisson's ratio is supposed to be computed in shear flexible beam elements, no difference is noted in the results for total strain up to 6.0%. The identical behavior is experienced for both Poisson's ratios when either pipe or elbow element models are used.

The paper investigates the effect of the element length on the resulting bending moment–strain relationship of the elbow element models. The study includes an assessment of the ELBOW31B element with only 3 ovality modes using five different element lengths (1D, 0.5 m, 1.0 m, 1.5 m, and 2.0 m), results are compared to PIPE31 element models with the same element lengths and are shown in Fig. 13.

As observed, Figs. 13 and 14 indicate that changing the element length either for elbow or pipe elements has an insignificant effect on the bending moment–strain relationship for single-element models. The sensitivity concludes that the 1D-element length is deemed appropriate for pure bending application, accordingly, reducing the time required to perform the analysis and postprocessing the results.

This section investigates the effect of increasing the number of elements of the ELBOW31 and the ELBOW31B models. All models between one element and ten elements this assessment consider a fixed element length of 323.85 mm as well as three ovality modes while the Poisson ratio is set to zero for all cases.

Figure 15 shows the results of the ELBOW31 model when NOWARP only, and when combined with NOOVAL restrictions are applied to both model ends. Results show good agreement for the ELBOW31B models when both 1 and 10 elements are used; it is also found to be matching both the pipe and shell element models until 1%.

On the other side, ELBOW31 presents an overpredicted behavior when a single element model is utilized with the application of both NOWARP and NOOVAL restrictions to the ends, while the 10 elements model with the same restrictions produces acceptable results to the desired total strain, i.e., 1%.

For the 10 elements ELBOW31 model, when NOOVAL and NOWARP were applied at the model ends, the ovalization at the center is increased therefore the bending moment–strain curve becomes closer to the Shell model. Due to the excellent similarities, comparison among the ELBOW31B element, ELBOW31 element with NOWARP only, with NOWARP and NOOVAL conditions, Shell and Pipe element results are the same as what described in the previous sections.

For the ELBOW31 with 10-elements, Fig. 16 shows the bending moment–strain relationship when the NOWARP only and the NOWARP and NOOVAL restrictions are applied to all model nodes. As it can be seen, when the NOWARP condition is applied on all nodes, the model produces the same results as the ten-element model with NOWARP at the model ends.

On the other hand, when ovalization restriction is introduced to all model nodes (i.e., NOOVAL is added to NOWARP condition), results present identical stiffening behavior to the single elbow element with zero ovality mode (Figs. 16 and 17), as well as the results with the NOWARP & NOOVAL conditions (Fig. 15) compared to the Shell and Pipe element models. As previously discussed, this overestimation is not consistent with the pipe bending analysis. Figure 17 below shows cases, which present an overpredicted bending moment–strain response for elbow elements.

Ovality, or the degree to which a circular cross section becomes elliptical, has a significant impact on the bending moment capacity of the pipelines. When a pipe or tubular structure undergoes bending, its circular cross section tends to deform into an elliptical shape due to the stresses and strains induced by the bending moment. This deformation “ovalization” reduces the structural capacity to carry additional bending loads and can lead to premature failure.

The current section presents the results of a study performed to investigate the effect of different ovality modes on the elbow element's ovalization. The study aims to show a comparison of the resulting ovalities to the SHELL element model, the equation depicted from the international standard PD 8010-2 [32], while ELBOW31B single-element type cases are used in this comparison and zero Poisson ratio.

where $f$ is the ovality; D_max, D_min, and D_nom are the maximum, minimum, and nominal pipe diameters, respectively. In this analysis, the outer diameter (OD = 323.85 mm) is the nominal diameter, Fig. 18 explains the ovality responses obtained for ELBOW31B element with different numbers of ovality modes, along with the 3D SHELL model results.

As observed, there is a significant difference between the pipe ovality obtained for one ovality mode when compared to the Shell element model. Ovality improves when ovality modes increase, i.e., when ovality modes are equal to or higher than 2, results show a good agreement to the Shell element model until a total strain of 1.0% (which is considered sufficient for most pipeline design applications). Although ovality modes 2 and 3 give accepted results until 1% strain, results are found to be underestimated for larger strain values; hence, higher ovality modes will be required. Ovality modes 4, 5, and 6 produce remarkably similar ovality responses when compared to the Shell element model for up to total strain of 2.5% strain. Beyond this strain level, the elbow element produces higher ovality results (with the six ovality mode case model producing the highest ovality values).

where $f$ and $f0$ are the ovality and the initial ovality, respectively; and D_nom and t_nom are the nominal diameter and the nominal wall thickness of the pipe, respectively. Parameter ε_b is the bending strain. The nominal wall thickness t_nom is the pipe wall thickness (9.525 mm).

The $Cp$ parameter is the coefficient that considers the effect of the external pressure. Thus, for pipes under pure bending (zero external pressure) and zero initial ovality, $Cp$ = 1, and $f0$ = 0.

The ovality equation referenced in PD 8010-2 [32] is based on elastic analysis. This means that the equation is applicable under conditions where the material behavior remains within the elastic range, ensuring that the pipeline returns to its original shape after the removal of the load. In this study, the results were analyzed with the understanding that the structural integrity of the pipeline must be maintained within these elastic limits, particularly during bending where ovalization can significantly affect the pipeline's performance. The findings presented here are consistent with the assumptions of the PD 8010 standard, ensuring compliance with the established design codes.

Figure 19 shows a comparison of the ovality obtained numerically (ELBOW31B model, and Shell model) and the ovality predicted by Eq. (2) (analytically). The agreement between the ovality results from Eq. (2) and the numerical models is seen to be remarkably good up to a total strain of 3.0%. At higher strains, ovalization results from PD 8010-2 [32] are underestimated.

Upon closer examination of the low strain area (<2%, as shown in Fig. 20), it is observed that the equation [32] underestimates ovalities. However, the difference is minimal, making the analytical solution still comparable.

The numerical analysis results are compared to the equation indicated in PD 8010-2, revealing insights into the limitations of the standard when applied to real scenarios. The findings suggest that while the standard's assumptions hold true under certain conditions, deviations occur when the material approaches its elastic limits. This indicates that the standard may require refinement in situations where the pipeline experiences significant bending strains.

The paper investigates the behavior of the ELBOW31 and ELBOW31B element subjected to pure bending using pipe elements and shell element models for benchmarking. The study begins by evaluating different model lengths to determine the suitable shell element model length. It is found that the bending moment–strain relationship remains consistent for strains up to 3.5%, regardless of the model length. However, a model length equal to six pipe diameters shows a larger bending moment when the total strain exceeds 3.5%, indicating a hardening effect. The study suggests that a total strain up to 1% is sufficient for pipeline design, aligning with industry standards.

For the ELBOW31 element, with no restrictions applied, the single element model shows acceptable results with 1 and 2 ovality modes, while additional ovality modes reduce the elastic bending moment capacity, making them less desirable. The introduction of the NOWARP end condition improves the response, providing acceptable results even with multiple ovality modes. Using 0 ovality mode, with or without NOWARP, overpredicts the elastic bending moment capacity and is thus not recommended.

The NOOVAL boundary condition alone yields unrealistic results for the ELBOW31 model, indicating inability of usage without further modifications. When both NOWARP and NOOVAL conditions are applied at the end nodes, the model similarly overpredicts the elastic capacity, showing little variation across different ovality modes.

The simplified ELBOW31B element shows good agreement with the ELBOW31-NOWARP model. However, the ELBOW31B model overpredicts the elastic capacity when zero ovality mode is used which is also not recommended.

The study finds no significant difference when different Poisson's ratios are used for all element types studied (ELBOW31, ELBOW31B, and PIPE31). Additionally, sensitivity analyses using different model lengths show no impact on the bending moment–strain response when no restrictions are applied.

Results show consistent bending moment–strain response when the NOWARP condition is applied at the model ends only, with no differentiation noticed when one-element or ten-element models are used for either ELBOW31 or ELBOW31B elements. Applying both NOWARP and NOOVAL conditions across all nodes in the ten-element ELBOW31 model leads to an overpredicted response, also when using only one element.

The ovality analysis shows that for strains up to 1%, the ELBOW31B element with more than one ovality mode closely matches the shell model. At higher strains, while two and three ovality modes show reduced ovality values, four to six modes provide a good match for strains up to 2.5%. Beyond this, five and six modes result higher ovality. Using only one ovality mode underpredicts ovality, leading to a stiffer bending moment–strain response.

The findings from this study, particularly those related to ovality and bending moment capacity have significant implications on the pipeline design as guided by standards like PD 8010. Selecting the appropriate element type based on these findings can ensure that pipelines are designed to withstand operational stresses, particularly in curved sections where ovality is a critical factor.

In addition, the study extends the analysis to pipes with D/t ratios of 20 and 50, with detailed results provided in Appendices A and B. For the D/t = 20 case (Appendix  A), the pipe exhibits a stiffer response and reduced ovalization, making it suitable for high-stress environments such as deepwater installations. Conversely, the D/t = 50 pipe (Appendix  B) shows increased flexibility and higher ovalization, suggesting greater vulnerability to structural deformation under bending loads.

These findings emphasize the importance of selecting the appropriate element type, boundary conditions, and D/t ratios based on the specific pipeline configuration and operational demands. Further research should explore the combined effects of internal pressure and bending, as well as the impact of different material properties on pipeline behavior to develop more comprehensive and reliable design guidelines.

While pipe elements provide a practical solution for modeling long straight sections of pipelines, they fall short in capturing the complex stress states in curved sections. This is where elbow elements come into play, offering detailed modeling capabilities for these critical areas. On the other hand, shell elements, though computationally intensive, provide the most detailed stress analysis for complex geometries, making them indispensable for certain high-stress applications.

This study validates the effectiveness of the elbow elements in simulating pipeline behavior under pure bending, providing a critical baseline for extending the models to more complex scenarios. The findings also align with the assumptions of PD 8010-2, while highlighting areas where the standard may require revision to account for nonelastic behavior.

The findings underscore the importance of selecting the right element type based on the specific pipeline configuration and operational conditions. For scenarios where bending is a dominant load condition, such as during pipeline installation in deepwater environments, using elbow elements with multiple ovality modes provides a more accurate representation of the pipeline's mechanical response. These elements effectively simulate the complexities of real-world pipeline behavior, ensuring that the design remains robust under various loading conditions.

Furthermore, pure bending has been established as a fundamental baseline for validating computational models and simulations. The accuracy of these models can be confirmed by comparing the results of the finite element analysis under pure bending conditions to the theoretical predictions or experimental data. Once validated, these models can be confidently extended to more complex scenarios involving internal and external pressures, ensuring that the bending behavior is correctly captured.

Analysis of different D/t ratios, as presented in Appendices A and B, further illustrates the critical role of wall thickness in determining the pipeline's structural performance. The results in Appendix  A show that a D/t ratio of 20 offers a stiffer response with higher elastic bending moment capacity and reduced ovalization, making it particularly suitable for high-stress environments like deepwater installations. Conversely, the findings in Appendix  B demonstrate that a D/t ratio of 50, while offering material efficiency, results in increased flexibility and greater ovalization, potentially compromising structural integrity under similar bending loads.

These findings highlight the need for careful consideration of D/t ratios in pipeline design, balancing material efficiency with mechanical robustness to meet specific operational requirements. The selection of the appropriate D/t ratio should be informed by the expected loading conditions to ensure the pipeline's long-term reliability.

Future research should investigate the combined effects of bending and internal pressures, as well as the influence of different material properties, building on the baseline established by this study.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

In Appendix  A, the study investigates the bending response of a pipe with a diameter-to-thickness (D/t) ratio of 20, which has a thicker wall. This configuration leads to a stiffer structural response and high bending moment capacity, making it suitable for high-stress applications such as deepwater pipelines. Key results are illustrated across multiple figures, each highlighting the behavior under different conditions:

• Figure 21 presents the bending moment–strain curve, showing a largely linear response that extends over a significant strain range. This indicates that the pipe can handle high bending loads before yielding, exhibiting robust structural performance.

• Figure 22 illustrates the response under the NOWARP condition, which suppresses warping effects. The bending moment–strain curve remains consistent, indicating that the pipe's deformation profile is stable even when strain increases. This stability reinforces the model's suitability for environments where resistance to deformation is essential.

• Figure 23 introduces various ovality modes, each contributing to a slightly decreased bending moment capacity. This subtle reduction indicates that the pipe accommodates localized deformation, increasing flexibility without substantial compromise in strength.

• Figure 24 combines the NOWARP and NOOVAL conditions, limiting both warping and ovalization. This produces a controlled deformation response, ideal for applications that demand minimized shape distortion under bending.

• Figure 25 shows the response of the simplified ELBOW31B model, which aligns well with the full ELBOW31 model under moderate strains, confirming the ELBOW31B model's utility for cases where simpler modeling suffices.

• Figure 26 compares bending moment–strain responses for ELBOW31 with NOWARP and ELBOW31B models, showing agreement under specific ovality modes, especially at lower strains.

• Figure 27 shows the effect of Poisson's ratio on the bending response of elbow and pipe elements, illustrating that Poisson's ratio has minimal impact up to 6% strain.

• Figure 28 examines the effect of different element lengths on bending moment–strain relationships, concluding that single-element models are appropriate for bending analysis without loss of accuracy.

• Figure 29 depicts bending moment–strain responses for a 10-element elbow model with restrictions applied only at the ends, showing consistency in behavior across configurations.

• Figure 30 displays the bending moment–strain response for 10-element elbow models, demonstrating acceptable accuracy for strain values up to 1%.

• Figure 31 highlights cases where overpredicted bending moment–strain responses occurred, emphasizing model limitations under specific boundary conditions.

• Figure 32 details the ovalization (or cross-sectional deformation) of the D/t = 20 pipe. The same figure illustrates multiple ovality modes for ELBOW31B.

• Figure 33 compares the results between ELBOW31B and the analytical model, covering strains up to 4%.

• Figure 34 presents similar comparisons as Figure 33 but focuses on strains up to 2%.

The results in Appendix  A emphasize the importance of a thicker wall for applications where high resistance to deformation is necessary. The D/t = 20 configuration, with minimal ovalization and high bending moment capacity, is particularly well-suited to environments requiring structural stability under significant loads.

Appendix  B examines a pipe with a D/t ratio of 50, which has a thinner wall and consequently exhibits a more flexible response under bending. This configuration results in a lower elastic bending moment capacity and an earlier onset of plastic deformation, reflecting a tradeoff between flexibility and structural robustness. Key figures provide insight into how this thinner-walled pipe behaves under various conditions:

• Figure 35 displays the bending moment–strain curve, where the pipe's flexibility is evident through the earlier deviation from linearity. This lower bending moment capacity indicates that the pipe begins to deform plastically at lower loads, making it less suitable for high-stress applications.

• Figure 36 shows the bending response with the NOWARP condition applied. The curve deviates from linearity at a lower strain level compared to the D/t=20 case, reflecting the reduced resistance of the thinner wall to bending loads and its susceptibility to ovalization.

• Figure 37 introduces multiple ovality modes, each causing a further reduction in bending moment capacity. This demonstrates how increased ovalization in thinner-walled pipes impacts structural response, compromising strength and making it vulnerable to shape changes under bending.

• Figure 38 applies both the NOWARP and NOOVAL conditions, which constrains both warping and ovalization effects. This results in a stiffer bending response, though it may be overly conservative for cases where flexibility is advantageous.

• Figure 39 presents the ELBOW31B model's bending moment–strain response, which aligns with ELBOW31 under moderate strains, confirming its effectiveness in simulating thinner pipes with reasonable accuracy.

• Figure 40 compares ELBOW31 with NOWARP and ELBOW31B models, highlighting agreement in their bending responses at lower strains.

• Figure 41 examines the effect of Poisson's ratio on bending responses, showing no significant difference across element types, even at higher strains.

• Figure 42 displays the effect of element length on bending moment–strain responses for both pipe and elbow elements, confirming element length has minimal impact.

• Figure 43 depicts the bending moment–strain response for 10-element elbow models with end restrictions only, confirming acceptable performance up to the target strain.

• Figure 44 shows bending moment–strain responses for 10-element models with restrictions across all nodes, indicating stiffened responses unsuitable for thinner-walled configurations.

• Figure 45 highlights cases where overpredicted responses occurred, particularly in models with combined NOWARP and NOOVAL conditions.

• Figure 46 illustrates the significant ovalization observed in the D/t = 50 pipe. The same figure shows ELBOWB with multiple ovality modes.

• Figure 47 compares these results with those from the ELBOW31B model, both showing a high degree of ovalization that compromises the pipe's structural integrity.

• Figure 48 shows comparison between ELBOW31B and analytical equation up to 4% strain.

The findings in Appendix  B highlight the tradeoffs of using a higher D/t ratio. While the thinner wall (D/t = 50) provides material efficiency and flexibility, it also introduces increased ovalization and deformation under bending loads, which could impact performance. For applications with high bending stresses, thicker-walled configurations, as seen in Appendix  A, offer superior performance.