by Helen Dunne, University of Oxford
This paper was first published in the June 2016 issue of GE and is based on the presentation delivered by Helen Dunne as part of the 2016 Cooling Prize compeition.
Rectangular shallow foundations, termed mudmats, are commonly used in the offshore oil and gas industry to support pipeline end terminations and pipeline end manifolds. Here, they are subjected to large horizontal forces that also give rise to overturning and torsional moments. This paper focuses on analysing the capacity of a rigid rectangular mudmat resting on the surface of undrained clay, when realistic three dimensional loading combinations are applied. A semi-analytical limit analysis method is derived, with more complex loading arrangements analysed using finite element limit analysis. With this technique it is possible to obtain lower and upper bounds on the exact load factor, and automative adaptive remeshing is used to reduce the bound gap over successive iterations of the solution. Adaptive mesh refinement helps visualisation of the failure mechanism and relationships between loading combinations and failure mechanisms are discussed.
A typical subsea arrangement of risers and flowlines for the offshore oil and gas industry can be seen in figure 1. Shallow foundations, termed mudmats, are used to support subsea infrastructure such as pipe line end terminations and pipe line end manifolds. These connect pipelines through jumpers, which can be situated at a height above the mudmat, and are often not aligned along the plan view axes. The jumper experiences horizontal expansion and contraction forces from the pipeline, which can induce biaxial bending, torsion and combined horizontal loads on the mudmat (depending on the jumper arrangement). Failure of mudmats can be significantly influenced by torsional loading (McDonald et al, 2014). A horizontal force that is not applied along one of the plan view axes causes a torsional moment on the mudmat, while a horizontal force applied at a height above the mudmat causes an overturning moment.
Classical bearing capacity theory, stemming from the Brinch Hansen (1970) method is adopted in the DNV (1992) and ISO (2003) design codes. This approach focuses on vertical bearing capacity. It has been found to be inaccurate in the analysis of foundations where the primary loading is not vertical, and foundations on undrained clay where short-term detachment of the foundation is prevented through under-base skirts (Ukritchon et al, 1998; Bransby and Randolph, 1998). A common alternative is the use of three-dimensional finite element analysis (Yun et al, 2009; Nouri et al, 2014; Feng et al, 2014). This improves accuracy but can be time-consuming as a mesh refinement study is required to ensure convergence of the solution.
Offshore pipelines are subjected to increasing levels of temperature and pressure change, leading to larger loads for mudmats to resist. However, mudmat dimensions are restricted by the size of their installation vessels – increasing the importance of an accurate bearing capacity analysis. Carrying out both lower bound (LB) and upper bound (UB) finite element limit analysis (FELA) is advantageous as the accuracy of the solution is directly quantifiable. Visualisation of the critical failure mechanism is also useful in the design process.
In this paper, analytical calculations and 3D FELA are used to analyse the bearing capacity of surface mudmats subject to combined loading. The notation and sign conventions used in this study are shown in figure 2. The mudmat is modelled as a rigid body of breadth, B, along the x-axis and length, L, along the y-axis. The mudmat length to breadth aspect ratio is fixed at L/B = 2, as this is typically adopted in practice (Feng et al, 2014).
Combined horizontal and torsional loading (Fx, Fy, Mz) is considered first. As it is improbable that a mudmat would be loaded from outside its plan area, emphasis is given to horizontal loads generating feasible levels of torsion. The largest feasible level of torsion that can be applied to a mudmat occurs when a horizontal load is applied at its corner. Next, the paper considers horizontal loads applied at a height, h, above the mudmat, resulting in combined horizontal and moment loading (figure 3). Forces applied at an eccentricity from, and a height above, the mudmat centre result in complex five degree of freedom loading (Fx, Fy, Mx, My, Mz). This paper explores the interaction between loading components and examines the corresponding mudmat failure mechanisms.
2.0 Finite element limit analysis
All analyses were performed using OxLim, a FELA program developed at the University of Oxford. OxLim has been used extensively for the analysis of plane strain problems (Martin, 2011; Martin and White, 2012; Mana et al, 2013; Dunne et al, 2015), and recently for 3D problems (Martin et al, 2015).
When using OxLim to solve a 3D problem, the soil domain is first discretised into a mesh of tetrahedral elements with relevant boundary conditions specified. The program then sets up two separate constrained optimisation problems that together allow rigorous bracketing of the exact collapse load multiplier. For this study, the LB analyses used a piecewise linear stress field, and the UB analyses used a piecewise linear velocity field. The bracketing error associated with the bounds is ± (UB - LB) / (UB + LB) and the average of the bounds, (UB + LB) / 2, is used as the best estimate solution.
Undrained failure was assumed and the soil was modelled as a rigid–plastic von Mises material. This allowed both the LB and UB analyses to be cast as standard second-order cone programming problems, and solved using specialised software (Makrodimopoulos and Martin, 2006; 2007; Mosek Aps, 2014).
Adaptive mesh refinement was used (if required) to improve the bracketing of the exact load multiplier. An adaptivity algorithm based on the spatial variation of the maximum shear strain rate in the UB velocity field was implemented, and all mesh generation was performed using the open-source code TetGen (Si, 2013). Sheaves of “singularity facets” were attached to the edges of the footing base (figure 4). When these facets are forced into the mesh, the bounds converge much more rapidly when compared to using a completely unstructured mesh.
The von Mises yield strength in pure shear, k, was equated with the Tresca shear strength, su, such that the two criteria were matched for deformation in plane strain. This has been found to provide improved accuracy when compared with using the shear strength under triaxial conditions in the analysis of 3D foundations (Gourvenec et al, 2006). The value of k was assumed as homogeneous throughout the soil domain. The footing/soil interface was modelled as fully rough with unlimited tensile capacity. This is consistent with the assumption that sufficient interior skirts are present in the mudmat to fully confine a soil plug (Mana et al, 2013). The soil was modelled as weightless, as soil weight has no effect on the results for this problem.
3.1 Horizontal and torsional loading
A surface footing subject to combined horizontal and torsional loading was analysed using a semi-analytical method based on the LB and UB plasticity theorems, and using FELA. LB values for Fx, Fy and Mz were found by numerically integrating the shear stress components τx and τy that would be induced by footing rotation about a prescribed point (x0,y0), as shown in figure 5a:
where τx = su cosα, τy = su sinα, with cosα and sinα calculated as shown in figure 5.
It is acknowledged that this lower bound solution is not fully rigorous as it does not extend the interface stress field into the remainder of the semi-infinite soil domain. However, this is considered a formality.
The UB solution was formulated by equating the internal and external work rates when the footing rotates with a virtual angular velocity, ω, about (x0,y0), as shown in figure 5b:
with r calculated as shown in figure 5. The 3D failure surface obtained by solving these equations numerically for a range of rotation centres is shown in figure 6a. It should be noted that the failure surface is symmetrical about the xy, yz, and xz planes. The LB equations produced results which matched exactly with the UB equations, indicating an exact theoretical solution. The circular failure surface when Mz = 0 highlights that the capacity of the footing in this case is always equal to the interface shear capacity, irrespective of the relative magnitudes of Fx and Fy.
Figure 6b shows that excellent agreement was found between the semi-analytical solution and the results obtained using FELA. FELA results for horizontal loading (Mz = 0) solved to ± 0.01% error without the need for mesh refinement, while in pure torsion (Fx, Fy = 0) an unrefined mesh solved to ±2% error. Each analysis completed in less than 10 seconds, and the critical failure mechanism was always interface shearing between the footing and the soil. The feasible torsion limit increased from 0.48B2Lsu when θ = 0 (Fx) to its maximum value 0.5B2Lsu, corresponding to θ = tan-1 (B/L) (Fθ). At θ = 90° (Fy) the feasible torsion limit is 0.38B2Lsu.
3.2 Horizontal, torsional and moment loading
In figure 7, bearing capacities are plotted for a footing subject to horizontal forces applied at a range of eccentricities from, and heights above, the centre point. When h/B = 0 and e/B = 0 the bearing capacity is the same whether the footing is loaded by Fx, Fy or Fθ (corresponding to the circular failure locus when Mz = 0 in figure 6). When these forces are applied at h/B = 0.5, there is no reduction in capacity compared with h/B = 0, and failure still occurs though interface shearing between the footing and the soil. When Fx and Fθ are applied at h/B = 1 there is a reduction in capacity compared with the same forces applied at h/B = 0. When h/B = 2 the reduction in capacity is substantial for Fx and Fθ, but there is still no significant change in capacity for a footing subject to Fy at this height.
The reduction in capacity when the footing is subject to Fx and Fθ at h/B > 0.5 can be attributed to the increased overturning moment acting on the footing, which can be visualised in the failure mechanism. Figure 8 shows the failure mechanisms when a footing is subject to forces in various directions at a height h/B = 2. Fx and Fθ cause failure by “rolling” about the y-axis. Fy predominantly fails through translation in the y direction with very little rotation. This highlights the increased moment capacity associated with ”pitching” rotation about the x-axis.
The bearing capacity generally reduces as the eccentricity, e, increases due to the additional torsional moment acting on the footing. This is less apparent when the footing is loaded at h/B = 2 by Fx or Fθ, as in this case, failure is still dominated by rolling about the y-axis and not twisting about the z-axis. Failure mechanisms are shown in figure 9 for a footing loaded by a corner force Fθ . When Fθ is applied at h/B = 0 and e/B = 0, failure occurs through translation with no rotation. When Fθ is applied at increasing eccentricities, the footing fails through translation as well as increasing amounts of twisting. This corresponds to the steep reduction in capacity when 0 ≤ e/B ≤ 1.12 in figure 7c. A footing subject to Fθ at e/B = 0 and h/B = 2 fails through rotation about the y-axis. The failure mechanism does not change significantly as e/B increases to 0.5, which corresponds to no change in the bearing capacity when h/B = 2 and 0 ≤ e/B ≤ 0.5 in figure 7c. When e/B =1.12 twisting is visible in the failure mechanism, and the bearing capacity is reduced.
This study has focused on likely loading scenarios for offshore mudmats used to support pipeline end terminations and manifolds. Rigorous 3D bearing capacity analyses have been carried out using both analytical calculations and FELA.
Horizontal forces were applied to a mudmat at eccentricities from, and heights above, its centre. Particular attention was given to identifying the levels of torsion and overturning moment that may realistically arise from non-concentric loading applied within the envelope of the mudmat and the equipment that it supports.
Applying a horizontal force at an eccentricity from the footing centre generates a torsional load component. The results quantify the gradual reduction in bearing capacity as the eccentricity of the horizontal force increases. The critical failure mechanism in this case changes from pure translation to combined translation and twisting.
A horizontal force applied at a height above the footing causes no reduction in bearing capacity until a certain threshold is reached, whereupon the capacity reduces rapidly (figure 7). This threshold was found to be higher when the force is aligned parallel to the longer side of the footing. The critical failure mechanism changes from pure translation to combined translation and rotation once the threshold has been reached.
The 3D FELA with adaptive mesh refinement was found to be an effective method for analysing the bearing capacity of shallow foundations on undrained clay subject to complex 3D loading. FELA results for a footing subject to combined horizontal and torsional loading showed excellent agreement with results from a semi-analytical solution. Combining LB and UB FELA proved beneficial as the bracketing error was often sufficiently small after the first iteration of the solution that mesh refinement was not necessary.
This research is part of a more extensive ongoing study aimed at optimizing the design and sizing of offshore mudmat foundations. Aspects such as footing embedment, footing/soil inter-face properties, vertical loading and soil weight are also being considered in this work.
The author is grateful for financial support received from Subsea 7 for her DPhil, and for helpful guidance and insight from her DPhil supervisor Chris Martin.
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